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Schauder and $L^{p}$ Elliptic Estimates

1 · Prerequisites

2 · Summary

This page develops the two classical second-order regularity theories for uniformly elliptic equations in nondivergence form and their interaction: the Schauder scale $C^{2, α}$ and the Sobolev scale $W^{2, p}$ .

The page begins with the H"older calculus: the spaces $C^{k, α}$ , their closure and boundary-extension classes, the scaled interior norms on which the estimates are stated, and the completeness of the closure class on a bounded domain. The cancelled representation of the second derivatives of Newtonian potentials supplies the model estimates for the Laplacian, and the freezing lemma together with the H"older interpolation $ε$ -loss produces the interior Schauder estimate for uniformly elliptic nondivergence operators. Boundary flattening, the quoted boundary regularity theorem and the half-space a priori estimate give the boundary Schauder estimate, assembled over a finite chart atlas. The boundary inputs are used under the Axiom of Choice.

In parallel, the Sobolev theory is built from the whole-space $W^{2, p}$ estimate for the Laplacian, the interpolation that absorbs first derivatives, the cutoff commutator, and the interior and global $W^{2, p}$ estimates on $C^{1, 1}$ domains; the odd-reflection half-space estimate is the boundary model. The two scales meet in the weak-to-strong regularity theorem for the Dirichlet Laplacian, in the classical solvability theorem for the Dirichlet problem obtained by the method of continuity, and in the comparison remark that the scales are not interchangeable; injectivity removes the $L^{p}$ kernel term from the global estimate, and the compactness-free method-of-continuity theorem supplies the abstract propagation step used in the solvability theorem.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains

Definition

Let $n \geq 1$ be an integer, let $k \geq 0$ be an integer, let $0 < α < 1$ and let $K \in {R, C}$ . Fix an open set $Ω \subseteq R^{n}$ and a function $u \in C^{k} (Ω; K)$ , where $C^{k}$ and the canonical-order derivatives $D^{β} u$ are those of $C^{k}$ maps and multi-index derivative notation in Euclidean space. Put $[u]_{k, α; Ω} : = \sum_{∣ β ∣ = k} \sup_{\begin{array}{c} x, y \in Ω \\ x \neq y \end{array}} \frac{∣ D^{β} u (x) - D^{β} u (y) ∣}{∣ x - y ∣^{α}}, ∥ u ∥_{C^{k, α} (Ω)} : = \sum_{j = 0}^{k} \sup_{x \in Ω} \max_{∣ β ∣ = j} ∣ D^{β} u (x) ∣ + [u]_{k, α; Ω},$ where the sums and maxima run over the finitely many multi-indices of the stated order. The quantity $[u]_{k, α; Ω}$ is the $k$ -th order $α$ -Hölder seminorm of $u$ and the quantity $∥ u ∥_{C^{k, α} (Ω)}$ the $C^{k, α}$ norm of $u$ ; both are taken in $[0, + \infty]$ , so the "norm" may be $+ \infty$ and only the class below carries a genuine normed-space structure.

The local Hölder class $C_{l o c}^{k, α} (Ω; K)$ consists of the $u \in C^{k} (Ω; K)$ with $[u]_{k, α; Ω^{'}} < + \infty$ for every $Ω^{'} ⋐ Ω$ . The bounded class $C^{k, α} (Ω; K) = C_{b}^{k, α} (Ω; K)$ consists of the $u \in C^{k} (Ω; K)$ with $∥ u ∥_{C^{k, α} (Ω)} < + \infty$ . Thus $C^{k, α} (Ω; K) \subseteq C_{l o c}^{k, α} (Ω; K)$ , and the inclusion may be strict: on $Ω = R^{n}$ the function $u (x) = \sin (∣ x ∣^{2})$ lies in the local class (it is $C^{1}$ , hence locally $α$ -Hölder on every compact set), while $[u]_{0, α; R^{n}} = + \infty$ , because at the points $x_{j} = \sqrt{2 π j} e_{1}$ and $y_{j} = x_{j} + (2 \sqrt{2 π j})^{- 1} e_{1}$ the difference quotients are $≍ (2 \sqrt{2 π j})^{α} \to \infty$ . On the bounded class the displayed formula is a norm, and the assignment $u \mapsto ∥ u ∥_{C^{k, α} (Ω)}$ is that norm.

Scaled interior norm. For a ball $B_{R} (x_{0}) \subseteq Ω$ of radius $R > 0$ and centre $x_{0}$ , write $[v]_{0, α; B} : = \sup {∣ v (x) - v (y) ∣ / ∣ x - y ∣^{α} : x, y \in B, x \neq y}$ for the plain Hölder seminorm on a set $B$ , and put $∥ u ∥_{k, α; B_{R} (x_{0})}^{*} : = \sum_{j = 0}^{k} R^{j} \max_{∣ β ∣ = j} \sup_{B_{R} (x_{0})} ∣ D^{β} u ∣ + R^{k + α} \max_{∣ β ∣ = k} [D^{β} u]_{0, α; B_{R} (x_{0})} .$ On every ball, $C^{2, α} (B_{R} (x_{0}))$ therefore agrees with the finite-scaled-norm class of that dependency. This is the scaled interior norm of $u$ on the ball; for $k = 2$ it is exactly the scaled quantity of Local Hölder and scaled C-two-alpha norms on balls, and it takes values in $[0, + \infty]$ as well. It is read off the open ball alone.

Bounded $C^{k, α}$ domains. A bounded $C^{k, α}$ domain in $R^{n}$ is a bounded nonempty open set $Ω \subseteq R^{n}$ with the following local graph property: for every $x \in \partial Ω$ there are an open neighbourhood $W$ of $x$ , a rigid motion $R (p) = Q p + b$ with orthogonal $Q$ and $b \in R^{n}$ , an open ball $B \subseteq R^{n - 1}$ and a function $φ \in C^{k, α} (B; R)$ such that, after shrinking $W$ so that $R (W) \subseteq B \times R$ , $R (Ω \cap W) = R (W) \cap {y = (y^{'}, y_{n}) \in B \times R : y_{n} < φ (y^{'})} .$ This generalises the integer-order notion of Bounded C^k domains and boundary charts to the Hölder scale, with the same one-sided graph convention; the regularity $φ \in C^{k, α} (B)$ is the only strengthening, connectedness is not required, and no boundary seminorm is attached to the interior norms above. In dimension $n = 1$ the corresponding sets are finite disjoint unions of bounded open intervals.

The boundary-extension class. Let $Ω$ be bounded and let $u \in C_{b}^{k, α} (Ω; K)$ . Say that $u$ lies in $C^{k, α} (\overline{Ω}; K)$ when each derivative field $D^{β} u$ with $∣ β ∣ \leq k$ extends continuously to $\overline{Ω}$ . The extension of each field is then unique, since $Ω$ is dense in $\overline{Ω}$ , and the sup and Hölder quantities formed with the extended fields and suprema over $\overline{Ω}$ agree with those displayed above, formed over $Ω$ : a supremum over the dense subset $Ω$ already computes the supremum of the extension, and for the seminorm the inequality $[\tilde{v}]_{0, α; \overline{Ω}} \leq [v]_{0, α; Ω}$ follows by approximating a pair in $\overline{Ω}$ by pairs in $Ω$ , so the two seminorms are equal. We equip $C^{k, α} (\overline{Ω}; K)$ with this common norm. This is the Hölder-scale analogue of the interior-up-to-boundary convention for integer order $C^{m} (\overline{Ω})$ fixed in Bounded C1 domains and their outward normals.

Remarks

Scaling. If $R > 0$ , $x_{0} \in R^{n}$ and $v (z) : = u (x_{0} + R z)$ on $B_{1} (0)$ , then $D^{β} v (z) = R^{∣ β ∣} D^{β} u (x_{0} + R z)$ , and consequently $∥ v ∥_{k, α; B_{1} (0)}^{*} = ∥ u ∥_{k, α; B_{R} (x_{0})}^{*}$ . The powers $R^{j}$ and $R^{k + α}$ are exactly what makes this identity hold; the same computation with $k = 2$ is the one recorded in Local Hölder and scaled C-two-alpha norms on balls.
Boundary extension. For $k = 0$ , every element of $C_{b}^{0, α} (Ω)$ is uniformly continuous and extends uniquely to $\overline{Ω}$ : for any boundary point choose an interior sequence converging to it, use the Hölder bound to make its values Cauchy, and compare two sequences by the same bound. For $k \geq 1$ , boundedness of the lower-order fields need not give their boundary limits on an arbitrary open set. For example, let $Ω = ((0, 1) \times (0, 2)) \cup ((1, 2) \times (0, 2))$ and let $u$ equal $0$ on the first component and $1$ on the second. All positive-order derivatives vanish, so $∥ u ∥_{C^{k, α} (Ω)} = 1$ for $k \geq 1$ , but $u$ has no limit at $(1, 1)$ . This set fails the one-sided boundary graph condition at the removed interface. The definitions assert no trace theorem, completeness or compactness.
Local versus bounded. The local class tests compactly contained subsets; the bounded class additionally controls all derivative suprema and the global top-order seminorm. On arbitrary bounded open sets, lower-order suprema can also fail: on the disjoint intervals $I_{j} = (2^{- j}, 2^{- j} + 2^{- j - 2})$ , the function equal to $j$ on $I_{j}$ is locally smooth with every positive-order derivative zero, but is unbounded. Thus boundedness of the domain alone does not identify the two classes.
Choice. The definition itself uses no choice principle; later completeness, approximation and embedding statements on this page state their own Countable Choice hypotheses.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

The closure Hölder spaces are Banach spaces

Statement

Assume Countable Choice. Let $n \geq 1$ , let $k \geq 0$ be an integer, $0 < α < 1$ , $K \in {R, C}$ and let $Ω \subseteq R^{n}$ be open and nonempty. Then $C_{b}^{k, α} (Ω; K)$ with the norm of Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains is a Banach space over $K$ : every Cauchy sequence in $∥ \cdot ∥_{C^{k, α} (Ω)}$ converges in that norm to a limit whose $k$ -th partial derivatives are $α$ -Hölder on $Ω$ . If in addition $Ω$ is a bounded $C^{k, α}$ domain with nonempty boundary, then the subspace $X_{0} : = {u \in C_{b}^{k, α} (Ω) : u extends continuously to \overset{ˉ}{Ω} with u ∣_{\partial Ω} = 0}$ is closed in $C_{b}^{k, α} (Ω)$ and hence is a Banach space. For every bounded open nonempty $Ω$ , the boundary-extension class $C^{k, α} (\overset{ˉ}{Ω}; K)$ is also a closed subspace of $C_{b}^{k, α} (Ω; K)$ and hence a Banach space, as is its zero-boundary subspace. Completeness here is for the full finite Hölder norm. A boundary Hölder seminorm alone does not define a norm on this function space, since it vanishes on nonzero constant functions.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , integers $k \geq 0$ , $0 < α < 1$ , $K \in {R, C}$ , an open nonempty $Ω \subseteq R^{n}$ , and a Cauchy sequence $(u_{j})$ in $C_{b}^{k, α} (Ω; K)$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ , used through the sequential completeness of $R$ and $C$ and the cited metric-space completeness conventions. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The norm is $∥ u ∥_{C^{k, α} (Ω)} = \sum_{j = 0}^{k} \sup_{Ω} \max_{∣ β ∣ = j} ∣ D^{β} u ∣ + [u]_{k, α; Ω}$ with $[u]_{k, α; Ω} = \sum_{∣ β ∣ = k} [D^{β} u]_{0, α; Ω}$ , and $C_{b}^{k, α}$ consists of the $C^{k}$ functions with finite norm; $D^{β}$ are the canonical-order derivatives. (Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains, $C^{k}$ maps and multi-index derivative notation in Euclidean space)

[F2]

Uniformly Cauchy sequences of real- (or complex-) valued functions converge uniformly; a uniform limit of continuous functions is continuous; and if $g_{m} \to g$ uniformly on an interval and the derivatives $g_{m}^{'}$ converge uniformly with $g_{m} (t_{0}) \to g (t_{0})$ at one point, then $g^{'} = \lim_{m} g_{m}^{'}$ . (A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy, The uniform limit of continuous real-valued functions on a metric space is continuous, A uniform limit of continuous complex-valued functions is continuous, If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit)

[F3]

A Banach space is a complete normed space; a subspace of a complete metric space is complete if and only if it is closed, under Countable Choice. (Banach space, Complete metric space: every Cauchy sequence converges in the space, Closed subspaces of complete metric spaces are complete; the converse under countable choice)

[F4]

The real mean value theorem bounds the increment of a differentiable function on a segment by the supremum of its derivative times the length of the segment. (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ )

Proof

technique · direct

1.1givenF1F2algebraA1

Uniform limits of the derivative fields. Since $(u_{j})$ is Cauchy in $∥ \cdot ∥_{C^{k, α} (Ω)}$ , for every multi-index $β$ with $∣ β ∣ \leq k$ the sequence $(D^{β} u_{j})$ is uniformly Cauchy on $Ω$ : for $j, l$ and every $x \in Ω$ , $∣ D^{β} u_{j} (x) - D^{β} u_{l} (x) ∣ \leq ∥ u_{j} - u_{l} ∥_{C^{k, α} (Ω)}$ . By [F2] there is a bounded function $v_{β}$ with $D^{β} u_{j} \to v_{β}$ uniformly on $Ω$ , and $v_{β}$ is continuous. Moreover $\sup_{Ω} ∣ v_{β} ∣ = \lim_{j} \sup_{Ω} ∣ D^{β} u_{j} ∣ \leq {lim inf}_{j} ∥ u_{j} ∥_{C^{k, α} (Ω)} < \infty$ , the last bound holding because a Cauchy sequence is bounded.

2.1step 1.1F1algebra

The limits are Hölder. For $∣ β ∣ = k$ and $x \neq y$ in $Ω$ , $∣ v_{β} (x) - v_{β} (y) ∣ = \lim_{j} ∣ D^{β} u_{j} (x) - D^{β} u_{j} (y) ∣ \leq {lim inf}_{j} [D^{β} u_{j}]_{0, α; Ω} ∣ x - y ∣^{α}$ ; hence $[v_{β}]_{0, α; Ω} \leq {lim inf}_{j} [D^{β} u_{j}]_{0, α; Ω} \leq {lim inf}_{j} ∥ u_{j} ∥_{C^{k, α} (Ω)} < \infty$ and $v_{β}$ is $α$ -Hölder on $Ω$ . Consequently $v_{0}$ (whose finiteness and continuity is step 1.1, including $k = 0$ , where no derivative is involved) satisfies $∥ v_{0} ∥_{C^{k, α} (Ω)} \leq {lim inf}_{j} ∥ u_{j} ∥_{C^{k, α} (Ω)} < \infty$ as soon as $v_{β} = D^{β} v_{0}$ for all $∣ β ∣ \leq k$ , which is proved next.

2.2step 1.1F1F2algebraF4

Identification of the limits with the derivatives of $v_{0}$ . Proceed by induction on $∣ β ∣$ . For $β = 0$ , $v_{0}$ is the limit. Suppose $v_{β} = D^{β} v_{0}$ is known on $Ω$ for some $∣ β ∣ < k$ ; fix $i$ and a ball $B ⋐ Ω$ (every point of $Ω$ lies in such a ball). On $B$ , all $u_{j}$ are $C^{k}$ ; for orders at least two, Continuous mixed partials of order $k$ are invariant under permutations identifies their derivative words with the canonical-order fields. Thus $D^{β} u_{j} \to v_{β}$ uniformly while $D^{β + e_{i}} u_{j} \to v_{β + e_{i}}$ uniformly; by [F2] applied to the restrictions to each coordinate segment inside $B$ (as in the one-variable theorem on a closed interval, at a fixed base point where $D^{β} u_{j}$ converges), the limit $v_{β}$ is differentiable in direction $e_{i}$ with $\partial_{i} v_{β} = v_{β + e_{i}}$ on $B$ ; Since every point lies in such a ball, this gives $\partial_{i} D^{β} v_{0} = D^{β + e_{i}} v_{0} = v_{β + e_{i}}$ on $Ω$ .

3.1step 2.1step 2.2F1F2F3algebra

Convergence in the Hölder norm. Let $ε > 0$ and choose $J$ with $∥ u_{j} - u_{l} ∥_{C^{k, α} (Ω)} \leq ε$ for $j, l \geq J$ . Fixing $l \geq J$ and passing to the limit in the componentwise bounds of steps 1.1 and 2.1 gives $\sup_{Ω} ∣ v_{β} - D^{β} u_{l} ∣ \leq ε$ for all $∣ β ∣ \leq k$ and $[v_{β} - D^{β} u_{l}]_{0, α; Ω} \leq ε$ for $∣ β ∣ = k$ ; hence $∥ v_{0} - u_{l} ∥_{C^{k, α} (Ω)} \leq C_{k} ε$ for every $l \geq J$ with a dimensional factor $C_{k}$ . So the Cauchy sequence converges in the norm to $v_{0} \in C_{b}^{k, α} (Ω)$ , and $C_{b}^{k, α} (Ω; K)$ is complete: it is a Banach space over $K$ (the vector-space operations are the pointwise ones and the norm is by [F1]). The complex case follows from the real case applied to real and imaginary parts, using [F2]'s complex uniform limit statement.

4.1step 3.1F1F2F3algebra

The boundary-condition subspace. Assume now $Ω$ is a bounded $C^{k, α}$ domain with nonempty boundary, and let $(u_{j})$ be a sequence in $X_{0}$ converging to $u$ in $C_{b}^{k, α} (Ω)$ . Each $u_{j}$ has a continuous extension ${\overset{ˉ}{u}}_{j}$ to $\overset{ˉ}{Ω}$ with ${\overset{ˉ}{u}}_{j} = 0$ on $\partial Ω$ . Since $\overset{ˉ}{Ω}$ is compact and the extensions are uniformly Cauchy on the dense set $Ω$ , they are uniformly Cauchy on $\overset{ˉ}{Ω}$ (for $x \in \overset{ˉ}{Ω}$ and $y \in Ω$ near $x$ , $∣ {\overset{ˉ}{u}}_{j} (x) - {\overset{ˉ}{u}}_{l} (x) ∣ = \lim_{y \to x} ∣ {\overset{ˉ}{u}}_{j} (y) - {\overset{ˉ}{u}}_{l} (y) ∣ \leq \sup_{Ω} ∣ u_{j} - u_{l} ∣$ ); hence ${\overset{ˉ}{u}}_{j}$ converges uniformly on $\overset{ˉ}{Ω}$ to a continuous $\overset{ˉ}{u}$ with $\overset{ˉ}{u} ∣_{Ω} = u$ and $\overset{ˉ}{u} = 0$ on the closed set $\partial Ω$ , so $u \in X_{0}$ . Thus $X_{0}$ is closed in the Banach space $C_{b}^{k, α} (Ω)$ , and [F3] makes it complete, hence a Banach space.

5.1step 3.1F1F2F3algebra∎

The closure class. Let $Ω$ be any bounded open nonempty set and let $u_{j} \in C^{k, α} (\overset{ˉ}{Ω}; K)$ converge to $u$ in $C_{b}^{k, α} (Ω; K)$ . For every $∣ β ∣ \leq k$ , let $w_{j, β}$ be the continuous extension of $D^{β} u_{j}$ . Density gives $\sup_{\overset{ˉ}{Ω}} ∣ w_{j, β} - w_{l, β} ∣ = \sup_{Ω} ∣ D^{β} u_{j} - D^{β} u_{l} ∣$ , so [F2] yields a continuous uniform limit $w_{β}$ on $\overset{ˉ}{Ω}$ . Its restriction is $D^{β} u$ , by convergence in the full norm. Thus $u$ belongs to the boundary-extension class, which is a closed linear subspace of $C_{b}^{k, α} (Ω; K)$ and is Banach by step 3.1 and [F3], with the same norm by [F1]. Its zero-boundary subspace is closed because the uniform limit $w_{0}$ of extensions vanishing on $\partial Ω$ also vanishes there, hence is Banach as well. No identification of the closure class with all of $C_{b}^{k, α} (Ω)$ is required.

Remarks

The interior completeness assertion holds for arbitrary open nonempty $Ω$ . The closure-class assertion assumes boundedness to match its definition; its proof and the closedness of the zero-boundary subspace require no boundary regularity. On an arbitrary bounded open set the closure class can be a proper closed subspace of $C_{b}^{k, α} (Ω)$ when $k \geq 1$ .
The local class $C_{l o c}^{k, α} (Ω)$ may contain functions with infinite full-domain norm; the displayed norm defines a Banach space on its finite-norm class $C^{k, α} (Ω) = C_{b}^{k, α} (Ω)$ . No assertion about completeness for a boundary pseudometric is made.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

The cancelled representation of the second derivatives of Newtonian potentials

Statement

Assume Countable Choice. Let $n \geq 2$ and let $Φ$ be the fundamental solution normalised by $- Δ Φ = δ_{0}$ on $R^{n}$ (Fundamental solution for the positive operator minus Laplacian); for $z \neq 0$ put $k_{i j} (z) : = \partial_{i} \partial_{j} Φ (z)$ and write $V_{i j} f : = p . v . (k_{i j} * f)$ as in Calderón–Zygmund kernels and their associated operators. Then:

(i) $k_{i j}$ is smooth off $0$ , homogeneous of degree $- n$ , satisfies $∣ k_{i j} (z) ∣ \leq C_{n} ∣ z ∣^{- n}$ and $∣ \nabla k_{i j} (z) ∣ \leq C_{n} ∣ z ∣^{- n - 1}$ , and hence $∣ k_{i j} (z - h) - k_{i j} (z) ∣ \leq C_{n} ∣ h ∣ ∣ z ∣^{- n - 1}$ whenever $∣ z ∣ \geq 2 ∣ h ∣$ ; its mean over every centred sphere is zero, so it is a standard Hölder Calderón–Zygmund kernel in the sense of Standard (Hölder) Calderón–Zygmund kernels.

(ii) For every $f \in C_{c}^{0, α} (R^{n})$ with $0 < α \leq 1$ , $N f$ has classical second derivatives and, for every $x \in R^{n}$ , $\partial_{i} \partial_{j} N f (x) = p . v . ⁣ \int_{R^{n}} k_{i j} (x - y) f (y) d y - \frac{δ_{i j}}{n} f (x) = \int_{∣ x - y ∣ < 1} k_{i j} (x - y) (f (y) - f (x)) d y + \int_{∣ x - y ∣ \geq 1} k_{i j} (x - y) f (y) d y - \frac{δ_{i j}}{n} f (x) .$ Both displayed ordinary integrals converge absolutely: the first by Hölder continuity at the singularity, the second because it avoids the singularity and $f$ is compactly supported. The local subtraction is only over the unit ball; the globally subtracted integrand $k_{i j} (x - y) (f (y) - f (x))$ is not absolutely integrable over $R^{n}$ when $f (x) \neq 0$ , since $k_{i j} (r θ) = r^{- n} k_{i j} (θ)$ , its nonzero continuous angular factor has positive spherical $L^{1}$ norm, and $\int_{1}^{\infty} r^{- 1} d r = \infty$ , so no global Lebesgue integral replaces the principal value.

(iii) As a tempered distribution, $D_{i j} Φ = p . v . k_{i j} - \frac{δ_{i j}}{n} δ_{0}$ ; consequently, for every $f \in L_{c}^{\infty} (R^{n})$ , the distributional second derivative of $N f$ is given by the truncated pairings, $\partial_{i} \partial_{j} N f = p . v . (k_{i j} * f) - \frac{δ_{i j}}{n} f in D^{'} (R^{n}),$ that is, $⟨ \partial_{i} \partial_{j} N f, φ ⟩ = \lim_{ε ↓ 0} \iint_{∣ x - y ∣ \geq ε} k_{i j} (x - y) f (y) φ (x) d y d x - \frac{δ_{i j}}{n} \int f φ$ for every $φ \in C_{c}^{\infty} (R^{n})$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , an integer $n \geq 2$ , the fundamental solution $Φ$ with the profiles below, the kernel $k_{i j} = \partial_{i} \partial_{j} Φ$ on $R^{n} ∖ {0}$ , and a continuous compactly supported $f$ with finite global Hölder seminorm $[f]_{0, α; R^{n}} < \infty$ for a fixed $0 < α \leq 1$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the choice-qualified measure, polar, surface, divergence, potential and distribution interfaces cited below. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

For $n \geq 3$ , $Φ (x) = ∣ x ∣^{2 - n} / ((n - 2) ω_{n - 1})$ ; for $n = 2$ , $Φ (x) = - (2 π)^{- 1} \log ∣ x ∣$ , $x \neq 0$ ; $Φ$ is locally integrable on $R^{n}$ with the pole assigned arbitrarily. Here $ω_{n - 1} = ∣ S^{n - 1} ∣$ . (Fundamental solution for the positive operator minus Laplacian, Euclidean spheres and closed balls as subspaces of $R^{n}$ )

[F2]

On $(0, \infty)$ , $(s^{γ})^{'} = γ s^{γ - 1}$ for real $γ$ and $(\log s)^{'} = 1 / s$ ; the chain rule and the algebra of derivatives compute derivatives of the radial profiles and of products; and the mean value theorem bounds the increment of a differentiable function on a segment by the supremum of its derivative times the segment length. (Real powers for positive bases, with the zero-base positive-exponent convention, Continuity and derivatives of positive-base real powers, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , $C^{k}$ maps and multi-index derivative notation in Euclidean space)

[F3]

Polar coordinates: $\int_{R^{n}} g d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} g (r θ) r^{n - 1} d σ (θ) d r$ ; the surface measure $σ$ is preserved by orthogonal transformations and the map $θ \mapsto a + R θ$ multiplies it by $R^{n - 1}$ ; $σ (S^{n - 1}) = ω_{n - 1} = n ∣ B_{1} ∣$ . (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Agreement with the existing polar sphere measure)

[F4]

The divergence theorem $\int_{Ω} div F d x = \int_{\partial Ω} F \cdot ν d S$ holds for bounded $C^{1}$ Euclidean domains and $F \in C^{1} (\overline{Ω}; R^{n})$ , with $ν$ the outward normal. (Divergence on a bounded C1 Euclidean domain)

[F5]

A distribution on an open set with support contained in ${0}$ equals a finite combination $\sum_{∣ γ ∣ \leq m} c_{γ} \partial^{γ} δ_{0}$ ; in particular the functional $φ \mapsto φ (0)$ generates the one-dimensional space of distributions with support ${0}$ that are invariant under all such combinations of order zero, so a distribution pairing $φ$ to exactly $\pm (δ_{i j} / n) φ (0)$ is $\pm (δ_{i j} / n) δ_{0}$ . (Distributions supported at one point)

[F6]

For bounded compactly supported data the Newtonian integral $N f (x) = \int Φ (x - y) f (y) d y$ is absolutely finite at every $x$ and $N f$ is locally bounded. (Newtonian potential of compactly supported data, Bounded compact data give an everywhere finite Newtonian potential)

[F7]

If $0 < β < 1$ , $g \in C_{c}^{0, β} (R^{n})$ and $N g$ is its Newtonian potential, then $N g \in C^{2} (R^{n})$ , $- Δ N g = g$ pointwise, and for every $x$ and every $r > 0$ with $supp g \subset B_{r} (x)$ , $\partial_{i} \partial_{j} N g (x) = \int_{B_{r} (x)} \partial_{i} \partial_{j} Φ (x - y) (g (y) - g (x)) d y - \frac{δ_{i j}}{n} g (x) .$ (Hölder data give a classical Newtonian solution)

[F8]

Here, for $0 < α \leq 1$ , $C_{c}^{0, α} (R^{n})$ denotes the continuous compactly supported functions with finite global seminorm $[f]_{0, α; R^{n}} : = \sup_{x \neq y} ∣ f (x) - f (y) ∣ / ∣ x - y ∣^{α}$ , and $∥ f ∥_{C^{0, α}} : = \sup ∣ f ∣ + [f]_{0, α; R^{n}}$ . For $0 < α < 1$ this is the convention in Hölder data give a classical Newtonian solution; $α = 1$ means Lipschitz continuity.

[F9]

A Calderón–Zygmund kernel with constants $A_{1}, A_{2}$ satisfies the annular size condition $\sup_{R > 0} \int_{R \leq ∣ x ∣ \leq 2 R} ∣ k ∣ \leq A_{1}$ and Hörmander's condition $\sup_{y \neq 0} \int_{∣ x ∣ \geq 2 ∣ y ∣} ∣ k (x - y) - k (x) ∣ d x \leq A_{2}$ ; it is standard $δ$ -Hölder with constant $A_{2}^{'}$ when additionally $∣ k (x - y) - k (x) ∣ \leq A_{2}^{'} ∣ y ∣^{δ} / ∣ x ∣^{n + δ}$ for $∣ x ∣ \geq 2 ∣ y ∣ > 0$ . A principal-value distribution for $k$ is a tempered distribution agreeing with $k$ off the origin and obtained as a truncation limit over some sequence $δ_{j} ↓ 0$ . (Calderón–Zygmund kernels and their associated operators, Standard (Hölder) Calderón–Zygmund kernels)

[F10]

For a function on a product of $σ$ -finite measure spaces that is integrable with respect to the product measure, the iterated integrals exist and agree (Fubini). (Fubini's theorem for L^1 functions on a sigma-finite product)

Proof

technique · direct

1.1givenF1F2algebraA1

Put $r = ∣ z ∣$ for $z \neq 0$ . Differentiating the profiles of [F1] with the rules of [F2] gives, in both cases $n \geq 3$ and $n = 2$ , $\partial_{i} Φ (z) = - \frac{z_{i}}{ω_{n - 1} r^{n}}, k_{i j} (z) = \partial_{i} \partial_{j} Φ (z) = - \frac{1}{ω_{n - 1}} \cdot \frac{1}{r^{n}} (δ_{i j} - n \frac{z_{i} z_{j}}{r^{2}}) .$ Both formulas are smooth in $z \neq 0$ , and $k_{i j} (λ z) = λ^{- n} k_{i j} (z)$ for every $λ > 0$ , so $k_{i j}$ is homogeneous of degree $- n$ .

2.1step 1.1F2algebra

Estimates. Directly from step 1.1, $∣ k_{i j} (z) ∣ \leq (1 + n) ω_{n - 1}^{- 1} r^{- n}$ . Since $k_{i j}$ is smooth and homogeneous of degree $- n$ on $R^{n} ∖ {0}$ , each partial derivative $\partial_{l} k_{i j}$ is homogeneous of degree $- n - 1$ , and its supremum over the compact unit sphere is a finite constant $C_{n, j, l}$ , so $∣ \nabla k_{i j} (z) ∣ \leq C_{n} r^{- n - 1}$ . If $∣ z ∣ \geq 2 ∣ h ∣$ , every point of the segment from $z - h$ to $z$ has norm at least $r / 2$ , so the mean value theorem bound in [F2], applied along that segment, gives $∣ k_{i j} (z - h) - k_{i j} (z) ∣ \leq C_{n} ∣ h ∣ (r / 2)^{- n - 1}$ , which is the stated pointwise estimate with a constant doubled to the power $n + 1$ .

2.2step 1.1F3algebra

Zero spherical means. On the unit sphere $k_{i j} (θ) = - ω_{n - 1}^{- 1} (δ_{i j} - n θ_{i} θ_{j})$ . By [F3] the measure $σ$ is invariant under coordinate reflections, so $\int θ_{i} θ_{j} d σ = 0$ for $i \neq j$ ; it is also invariant under coordinate permutations, so all $n$ numbers $\int θ_{i}^{2} d σ$ are equal, and $\sum_{i} θ_{i}^{2} = 1$ gives $\int θ_{i}^{2} d σ = ω_{n - 1} / n$ . Hence $\int_{S^{n - 1}} θ_{i} θ_{j} d σ = δ_{i j} ω_{n - 1} / n$ and therefore $\int_{S^{n - 1}} k_{i j} (θ) d σ (θ) = - ω_{n - 1}^{- 1} (δ_{i j} ω_{n - 1} - n δ_{i j} ω_{n - 1} / n) = 0$ . By the homogeneity of step 1.1 the mean over every centred sphere vanishes: $\int_{S (0, ρ)} k_{i j} d S = ρ^{n - 1} ρ^{- n} \int_{S^{n - 1}} k_{i j} (θ) d σ = 0$ .

3.1step 2.1step 2.2F3F9algebra

$k_{i j}$ is a standard Hölder Calderón–Zygmund kernel. The annular size condition follows from step 2.1 and [F3]: $\int_{R \leq ∣ z ∣ \leq 2 R} ∣ k_{i j} ∣ d z = \int_{R}^{2 R} ρ^{- 1} d ρ \int_{S^{n - 1}} ∣ k_{i j} (θ) ∣ d σ \leq (1 + n) \log 2 = : A_{1}$ . Hörmander's condition follows from the difference estimate of step 2.1 and polar integration: for $y \neq 0$ , $\int_{∣ x ∣ \geq 2 ∣ y ∣} ∣ k_{i j} (x - y) - k_{i j} (x) ∣ d x \leq C_{n} ∣ y ∣ \int_{2 ∣ y ∣}^{\infty} ρ^{- n - 1} ω_{n - 1} ρ^{n - 1} d ρ = C_{n} ω_{n - 1} / 2 = : A_{2}$ . The pointwise estimate of step 2.1 is exactly condition (1) of [F9] with $δ = 1$ and constant $C_{n} 2^{n + 1}$ , so $k_{i j}$ is standard $1$ -Hölder in the sense of Standard (Hölder) Calderón–Zygmund kernels.

3.2step 2.2F3F9algebra

The principal value exists. Let $φ \in S (R^{n})$ and $0 < ε < 1$ . Using the vanishing spherical means of step 2.2 on the annulus, $\int_{∣ z ∣ \geq ε} k_{i j} (z) φ (z) d z = \int_{ε \leq ∣ z ∣ < 1} k_{i j} (z) (φ (z) - φ (0)) d z + \int_{∣ z ∣ \geq 1} k_{i j} (z) φ (z) d z .$ The second integral is absolutely convergent because $∣ k_{i j} (z) φ (z) ∣ \leq C_{n} ∣ z ∣^{- n} ∣ φ (z) ∣$ and $φ$ is Schwartz, and the first integrand is dominated by $C_{n} ∥ \nabla φ ∥_{\infty} ∣ z ∣^{1 - n}$ , which is integrable on $∣ z ∣ < 1$ for $n \geq 2$ ; hence the first integral converges as $ε ↓ 0$ by dominated convergence. Therefore $W_{i j} (φ) : = \lim_{ε ↓ 0} \int_{∣ z ∣ \geq ε} k_{i j} (z) φ (z) d z$ exists for every Schwartz $φ$ ; the bound $∣ W_{i j} (φ) ∣ \leq C_{n} (∥ \nabla φ ∥_{\infty} + \sup_{z} (1 + ∣ z ∣)^{2} ∣ φ (z) ∣)$ proves continuity in Schwartz seminorms, so the map is a tempered distribution agreeing with $k_{i j}$ off the origin, and taking $δ_{j} = 1 / j$ exhibits it as a principal-value distribution for $k_{i j}$ in the sense of [F9].

3.3step 2.2F7F8algebra

Classical formula for Hölder data. If $0 < α < 1$ put $β : = α$ , and if $α = 1$ put $β : = 1 / 2$ ; $f \in C_{c}^{0, β} (R^{n})$ in either case: for $α = 1$ , if $∣ x - y ∣ \leq 1$ then $∣ f (x) - f (y) ∣ / ∣ x - y ∣^{β} \leq [f]_{0, 1}$ , while if $∣ x - y ∣ > 1$ it is at most $2 ∥ f ∥_{\infty}$ . Thus $[f]_{0, β} \leq \max {[f]_{0, 1}, 2 ∥ f ∥_{\infty}} < \infty$ , so [F7] gives $N f \in C^{2} (R^{n})$ together with the ball formula. Fix $x \in R^{n}$ and choose $r > 1$ with $supp f \subset B_{r} (x)$ ; substituting $z = x - y$ and splitting ${∣ z ∣ < r}$ into ${∣ z ∣ < 1}$ and ${1 \leq ∣ z ∣ < r}$ gives $\int_{B_{r} (x)} k_{i j} (x - y) (f (y) - f (x)) d y = \int_{∣ x - y ∣ < 1} k_{i j} (x - y) (f (y) - f (x)) d y + \int_{∣ x - y ∣ \geq 1} k_{i j} (x - y) f (y) d y,$ because the pure multiple term $- f (x) \int_{1 \leq ∣ z ∣ < r} k_{i j} (z) d z$ vanishes by the annular zero-mean property of step 2.2, while the remaining ${1 \leq ∣ z ∣ < r}$ -integral equals the integral over ${∣ z ∣ \geq 1}$ : indeed $f (x + z) = 0$ for $∣ z ∣ \geq r$ , so the two extended integrands agree almost everywhere. Hence $\partial_{i} \partial_{j} N f (x) = \int_{∣ x - y ∣ < 1} k_{i j} (x - y) (f (y) - f (x)) d y + \int_{∣ x - y ∣ \geq 1} k_{i j} (x - y) f (y) d y - \frac{δ_{i j}}{n} f (x)$ for the fixed $x$ .

3.4step 2.2F3F8algebra

The principal value and the two integrals. With $0 < ε < 1$ , the vanishing of $\int_{ε \leq ∣ x - y ∣ < 1} k_{i j} (x - y) d y$ by step 2.2 gives $\int_{∣ x - y ∣ \geq ε} k_{i j} (x - y) f (y) d y = \int_{ε \leq ∣ x - y ∣ < 1} k_{i j} (x - y) (f (y) - f (x)) d y + \int_{∣ x - y ∣ \geq 1} k_{i j} (x - y) f (y) d y .$ The first integrand is bounded by $C_{n} [f]_{0, α; R^{n}} ∣ x - y ∣^{α - n}$ , whose integral over $∣ x - y ∣ < 1$ is finite because $α - n > - n$ , so dominated convergence lets $ε ↓ 0$ pass to the absolutely convergent first integral of the statement; the other integral is absolutely convergent because $∣ k_{i j} (x - y) f (y) ∣ \leq C_{n} ∣ f (y) ∣$ on $∣ x - y ∣ \geq 1$ and $f$ is integrable. Hence the principal value $p . v . \int k_{i j} (x - y) f (y) d y$ exists for every $x$ and equals the sum of the two integrals of the displayed formula.

4.1step 1.1step 2.2step 3.2F3F4F5algebra

Distributional identity. Let $φ \in C_{c}^{\infty} (R^{n})$ and choose $R$ with $supp φ \subset B_{R / 2} (0)$ . For $0 < ε < R / 2$ apply the divergence theorem [F4] on the bounded $C^{1}$ domain $Ω_{ε} : = {ε < ∣ x ∣ < R}$ to the field $F : = Φ \partial_{j} φ e_{i} - φ \partial_{i} Φ e_{j}$ . Since $div F = \partial_{i} (Φ \partial_{j} φ) - \partial_{j} (φ \partial_{i} Φ) = Φ \partial_{i} \partial_{j} φ - k_{i j} φ$ and $F = 0$ near $∣ x ∣ = R$ , while $ν = - x / ∣ x ∣$ on $∣ x ∣ = ε$ , $\int_{Ω_{ε}} Φ \partial_{i} \partial_{j} φ d x = \int_{Ω_{ε}} k_{i j} φ d x + \int_{∣ x ∣ = ε} (Φ \partial_{j} φ ν_{i} - φ \partial_{i} Φ ν_{j}) d S .$ On $∣ x ∣ = ε$ the first boundary term tends to $0$ as $ε ↓ 0$ , because $Φ = O (ε^{2 - n})$ or $O (\log (1 / ε))$ and $d S$ contributes $ε^{n - 1}$ ; the second boundary term is $- \int_{∣ x ∣ = ε} φ \partial_{i} Φ ν_{j} d S$ , and by step 1.1, the outward normal $ν = - x / ε$ and the scaling rule of [F3], $\int_{∣ x ∣ = ε} φ \partial_{i} Φ ν_{j} d S = \frac{1}{ω_{n - 1} ε^{n + 1}} \int_{∣ x ∣ = ε} φ (x) x_{i} x_{j} d S (x) ⟶ \frac{δ_{i j}}{n} φ (0),$ because $\int_{∣ x ∣ = ε} φ x_{i} x_{j} d S = ε^{n + 1} \int_{S^{n - 1}} φ (ε θ) θ_{i} θ_{j} d σ (θ)$ and $φ (ε θ) = φ (0) + O (ε)$ . Hence the boundary integral in the identity tends to $- \frac{δ_{i j}}{n} φ (0)$ . Passing to the limit, using local integrability of $Φ$ on the left and step 3.2 on the right, gives $⟨ \partial_{i} \partial_{j} Φ, φ ⟩ = ⟨ Φ, \partial_{i} \partial_{j} φ ⟩ = W_{i j} (φ) - \frac{δ_{i j}}{n} φ (0)$ ; by [F5] this is the distributional identity $D_{i j} Φ = p . v . k_{i j} - \frac{δ_{i j}}{n} δ_{0}$ .

5.1step 2.1step 2.2step 4.1step 3.3step 3.4F3given

Assembly. Combining steps 3.3 and 3.4 gives, for every $x \in R^{n}$ , $\partial_{i} \partial_{j} N f (x) = p . v . \int k_{i j} (x - y) f (y) d y - (δ_{i j} / n) f (x)$ , and the two ordinary integrals in the equivalent displayed form converge absolutely by step 3.4; step 3.3 supplies the classical second derivatives. The subtraction of $f (x)$ is made only on the unit ball: the globally subtracted integrand obeys $∣ k_{i j} (x - y) (f (y) - f (x)) ∣ \geq \frac{1}{2} ∣ f (x) ∣ ∣ k_{i j} (x - y) ∣$ for all $y$ outside a large ball when $f (x) \neq 0$ , and $\int_{∣ z ∣ \geq 1} ∣ k_{i j} (z) ∣ d z = \int_{1}^{\infty} ρ^{- 1} d ρ \int_{S^{n - 1}} ∣ k_{i j} (θ) ∣ d σ = \infty$ by [F3] and step 1.1, so no absolutely convergent Lebesgue integral over $R^{n}$ can replace the principal value; the annular cancellation of step 2.2 is what makes the truncations converge. The case $α = 1$ was reduced to $β = 1 / 2 < α$ in step 3.3, where higher Hölder regularity than assumed is irrelevant to the formula; the positivity of $α$ makes $∣ z ∣^{α - n}$ integrable at the origin in step 3.4. All constructions are pointwise in $x$ and use only Countable Choice in the cited measure, polar, potential and distribution interfaces.

6.1step 4.1F5F6F10algebra∎

Distributional form for bounded compact data. Let $f \in L_{c}^{\infty} (R^{n})$ and $φ \in C_{c}^{\infty} (R^{n})$ . The double integral $\iint Φ (x - y) f (y) \partial_{i} \partial_{j} φ (x) d y d x$ is absolutely convergent: the integrand is supported in a bounded subset of $R^{n} \times R^{n}$ and is dominated there by a constant times $∣ Φ (x - y) ∣$ , and $Φ$ is locally integrable. Fubini [F10] and the substitution $z = x - y$ therefore give $⟨ D_{i j} N f, φ ⟩ = \int f (y) ⟨ Φ, D_{i j} φ (\cdot + y) ⟩ d y = \int f (y) ⟨ D_{i j} Φ, φ (\cdot + y) ⟩ d y$ , the last equality being the definition of the distributional derivative. Inserting the kernel identity proved in step 4.1, $⟨ D_{i j} N f, φ ⟩ = \int f (y) [W_{i j} (φ (\cdot + y)) - \frac{δ_{i j}}{n} φ (y)] d y$ . For $y$ in the compact support of $f$ , subtract $φ (y)$ on $ε \leq ∣ z ∣ < 1$ using step 2.2. The resulting near integrand is bounded by $C_{n} ∥ \nabla φ ∥_{\infty} ∣ z ∣^{1 - n}$ ; the far integral is bounded by $C_{n} ∥ φ ∥_{\infty} \int_{1 \leq ∣ z ∣ \leq R_{0}} ∣ z ∣^{- n} d z$ , where one $R_{0} > 1$ contains all differences of the two supports. These bounds are uniform in $y$ and $ε$ ; multiplying by $∣ f (y) ∣$ gives an integrable majorant on its compact support. Dominated convergence now passes the truncation limit through the $y$ integral; another application of Fubini [F10] (the truncated double integral is absolutely convergent on the bounded region) gives $\int f (y) W_{i j} (φ (\cdot + y)) d y = \lim_{ε ↓ 0} \iint_{∣ x - y ∣ \geq ε} k_{i j} (x - y) f (y) φ (x) d y d x$ . Hence $⟨ \partial_{i} \partial_{j} N f, φ ⟩ = \lim_{ε ↓ 0} \iint_{∣ x - y ∣ \geq ε} k_{i j} (x - y) f (y) φ (x) d y d x - \frac{δ_{i j}}{n} \int f φ$ for every test function, which is clause (iii).

Remarks

The sign of the correction term is fixed by step 4.1 and checked against the trace: summing the identity over $i = j$ gives $Δ Φ = p . v . (Δ Φ) - δ_{0} = - δ_{0}$ off the origin, that is $- Δ Φ = δ_{0}$ , the normalisation of Fundamental solution for the positive operator minus Laplacian. This is a consistency check, not a substitute for step 4.1.
Nothing here asserts a global subtracted identity, and no smoothness of $f$ beyond continuity and the stated Hölder modulus is used; the formula for $N f$ is the same for real- or complex-valued data after applying the real case to real and imaginary parts.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Uniformly elliptic nondivergence-form operators and their frozen coefficients

Definition

Let $n \geq 1$ and let $Ω \subseteq R^{n}$ be open. A second-order nondivergence-form operator on $Ω$ is an expression $L u = a^{i j} \partial_{i} \partial_{j} u + b^{i} \partial_{i} u + c u,$ with real-valued bounded measurable coefficients $a^{i j}, b^{i}, c$ on $Ω$ and summation over the repeated indices $i, j \in {1, \dots, n}$ . In this operator notation, coordinates and coordinate partials are relabelled from $C^{k}$ maps and multi-index derivative notation in Euclidean space: coordinate $i$ and $\partial_{i}$ here mean coordinate $i - 1$ and $\partial_{i - 1}$ there, and likewise $ξ_{i}$ means component $i - 1$ of $ξ$ . Multi-index derivatives $D^{β}$ retain that dependency's zero-based canonical order. The expression acts on functions for which the displayed classical derivatives exist. The matrix field $A = (a^{i j})_{i, j = 1}^{n}$ is the principal coefficient matrix, and $(b^{i})$ and $c$ are the lower-order coefficients.

The operator is uniformly elliptic on $Ω$ with constants $0 < λ \leq Λ < \infty$ when $A (x)$ is symmetric for almost every $x \in Ω$ and $λ ∣ ξ ∣^{2} \leq a^{i j} (x) ξ_{i} ξ_{j} \leq Λ ∣ ξ ∣^{2}$ for every $ξ \in R^{n}$ and almost every $x \in Ω$ . The number $Λ / λ \geq 1$ is the ellipticity ratio; uniform ellipticity is a condition on the pointwise spectrum of $A$ .

For $x_{0} \in Ω$ the frozen operator at $x_{0}$ is the constant-coefficient operator $L_{x_{0}} : = a^{i j} (x_{0}) \partial_{i} \partial_{j}$ built from the principal matrix at the single point $x_{0}$ ; when the coefficients are continuous at $x_{0}$ the frozen operator is to be regarded as the constant-coefficient model of $L$ near $x_{0}$ . When the principal coefficients are of class $C^{0, α}$ with respect to Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains on a ball $B \subseteq Ω$ , one writes $[A]_{0, α; B} \leq K$ for the maximum over $i, j$ of the Hölder seminorms $[a^{i j}]_{0, α; B}$ , and when $∥ b ∥_{\infty} + ∥ c ∥_{\infty} \leq M$ on $B$ one says that the lower-order coefficients of $L$ are bounded by $M$ on $B$ .

Remarks

What is asserted. The definition fixes the coefficient classes, the sign-free ellipticity condition, the frozen-coefficient notation and the quantitative coefficient bounds. It asserts no solvability of $L u = f$ , no weak or distributional formulation, no continuity, Hölder or VMO regularity of the coefficients beyond what is explicitly stated, and no symmetry of the lower-order coefficients. Every estimate or solvability statement on this page states its own hypotheses on the coefficient regularity and on the data.
Two regimes. The Schauder theory on this page uses bounded principal coefficients with finite full-ball Hölder seminorm, quantitatively $[A]_{0, α; B} \leq K < \infty$ (membership in $C^{0, α} (B)$ under the finite-norm convention). Local membership in $C_{l o c}^{0, α} (B)$ alone does not imply this bound; the $W^{2, p}$ theory uses only the continuity of the principal coefficients on the closed ball. Both hypotheses appear separately in the statements below, and no estimate silently upgrades one to the other or to a VMO/measurable regime.
Frozen coefficients. If $A$ is continuous at $x_{0}$ , its almost-everywhere symmetry and ellipticity extend to $x_{0}$ : choose points outside the common null exceptional set tending to $x_{0}$ and pass to the limit in the matrix identities and quadratic inequalities. Then $L_{x_{0}}$ has the same ellipticity constants. For merely measurable coefficients, the value at an exceptional point can be changed arbitrarily, so this conclusion is unavailable there. If $A$ is continuous at $x_{0}$ then $A (x) \to A (x_{0})$ as $x \to x_{0}$ , which is the small-scale input used to absorb the oscillation of $A (x) - A (x_{0})$ .
Scale-normalized bounds. For a ball $B_{R} (x_{0})$ and a coefficient matrix in $C^{0, α}$ , the dimensionless quantities appearing in the estimates of this page are $R^{α} K$ , $R ∥ b ∥_{\infty}$ , $R^{1 + α} [b]_{0, α}$ , $R^{2} ∥ c ∥_{\infty}$ and $R^{2 + α} [c]_{0, α}$ ; the powers are those of the scaling of the corresponding derivative orders. No choice principle is used in this definition.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Ehrling-type Hölder and derivative interpolation with an epsilon loss

Statement

Let $n \geq 1$ , $0 < α < 1$ , let $k \geq 1$ be an integer, $R > 0$ , and let $B_{R} \subseteq Ω$ be a ball in an open set. For every $ε > 0$ there is $C = C (n, k, α, ε) < \infty$ such that every $u$ with $∥ u ∥_{k, α; B_{R}}^{*} < \infty$ satisfies $(i) \sum_{j = 0}^{k - 1} R^{j} \max_{∣ β ∣ = j} \sup_{B_{R}} ∣ D^{β} u ∣ \leq ε ∥ u ∥_{k, α; B_{R}}^{*} + C \sup_{B_{R}} ∣ u ∣,$ $(ii) \max_{∣ β ∣ = k} \sup_{B_{R}} ∣ D^{β} u ∣ \leq ε R^{α} \max_{∣ β ∣ = k} [D^{β} u]_{0, α; B_{R}} + C R^{- k} \sup_{B_{R}} ∣ u ∣ .$ For $0 < α < β \leq 1$ and $f \in C^{0, β} (B_{R})$ , one also has $(iii) [f]_{0, α; B_{R}} \leq ε R^{β - α} [f]_{0, β; B_{R}} + C (n, α, β) ε^{- α / (β - α)} R^{- α} \sup_{B_{R}} ∣ f ∣ .$ The constants are independent of $u, f, R$ ; the inequalities are scale-invariant.

Facts & Assumptions

Given: $n \geq 1$ , $0 < α < 1$ , an integer $k \geq 1$ , a radius $R > 0$ , a ball $B_{R} = B_{R} (x_{0})$ , a fixed $ε > 0$ , and a function $u$ with $∥ u ∥_{k, α; B_{R}}^{*} < \infty$ (respectively $f \in C^{0, β} (B_{R})$ in the third part).

[F1]

The scaled norm is $∥ u ∥_{k, α; B_{R}}^{*} = \sum_{j = 0}^{k} R^{j} \max_{∣ β ∣ = j} \sup_{B_{R}} ∣ D^{β} u ∣ + R^{k + α} \max_{∣ β ∣ = k} [D^{β} u]_{0, α; B_{R}}$ , and $[v]_{0, γ; B} = \sup_{B} ∣ v (x) - v (y) ∣ / ∣ x - y ∣^{γ}$ . Under $v (z) = u (x_{0} + R z)$ one has $D^{β} v (z) = R^{∣ β ∣} D^{β} u (x_{0} + R z)$ , the scaling identity $∥ v ∥_{k, α; B_{1}}^{*} = ∥ u ∥_{k, α; B_{R}}^{*}$ , and for $g (z) = f (x_{0} + R z)$ the identities $[g]_{0, γ; B_{1}} = R^{γ} [f]_{0, γ; B_{R}}$ for $γ \in {α, β}$ and $\sup_{B_{1}} ∣ g ∣ = \sup_{B_{R}} ∣ f ∣$ . (Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains, Local Hölder and scaled C-two-alpha norms on balls)

[F2]

All derivatives are canonical-order partial derivatives. The mean value theorem bounds increments along a segment, and iterating Botsko's theorem: if $F$ is continuous on $[a, b]$ , $F^{'} (x) = f (x)$ off a countable subset of $(a, b)$ , and $f$ is Riemann integrable, then $\int_{a}^{b} f = F (b) - F (a)$ on the smooth restrictions to a segment gives the Taylor formula with integral remainder; Continuous mixed partials of order $k$ are invariant under permutations identifies derivative words of orders at least two. Consequently $∣ v (x + h) - \sum_{∣ γ ∣ < m} \frac{D^{γ} v (x)}{γ!} h^{γ} ∣ \leq C_{n, m} ∣ h ∣^{m} \max_{∣ γ ∣ = m} \sup_{[x, x + h]} ∣ D^{γ} v ∣$ . Under a linear change $y = x + V z$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ expresses each $z$ -derivative of order $m$ as a linear combination of $y$ -derivatives of order $m$ , with coefficients bounded in terms of $m, n, ∥ V ∥$ ; if $V$ is invertible with bounded inverse, the same holds in reverse. An $α$ -Hölder bound for the top-order $y$ -derivatives therefore gives the corresponding $z$ -derivative bound, with a factor controlled by $∥ V ∥^{α}$ . (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{k}$ maps and multi-index derivative notation in Euclidean space)

[F3]

Young's inequality for real exponents: for conjugate exponents $p, q > 1$ and $a, b \geq 0$ one has $a b \leq ε a^{p} + C_{p, q} ε^{- q / p} b^{q}$ ; in part (ii) it is applied with $p = (k + α) / k$ and $q = (k + α) / α$ , both greater than one. Part (iii) uses only its direct low/high increment split and does not use Young's inequality. (Young's inequality for conjugate real exponents)

Proof

technique · direct

1.1F1given

Reduction to $B_{1}$ . Put $x_{0}$ for the centre of $B_{R}$ , $v (z) : = u (x_{0} + R z)$ on $B_{1}$ and $g (z) : = f (x_{0} + R z)$ . By the scaling identities of [F1] the three claims for $(u, f, R)$ are equivalent to the same claims for $(v, g, 1)$ : the factors $R^{j}$ , $R^{k + α}$ , $R^{α}$ , $R^{- k}$ and $R^{γ}$ reproduce exactly the displayed powers. It therefore suffices to prove all three statements for $R = 1$ with constants independent of the function; this is assumed from now on.

1.2F1givencasesalgebra

Part (iii). Let $0 < α < β \leq 1$ , $f \in C^{0, β} (B_{1})$ , $M : = \sup_{B_{1}} ∣ f ∣$ and $F : = [f]_{0, β; B_{1}}$ . If $M = 0$ or $F = 0$ the claim is immediate (for $F = 0$ , use that $f$ is constant, so $[f]_{0, α} = 0$ ; for $M = 0$ the function vanishes), so assume $M, F > 0$ . If $ε \geq 2^{β - α}$ , then for all $x \neq y$ in $B_{1}$ , $∣ x - y ∣ \leq 2$ and $[f]_{0, α; B_{1}} \leq F \cdot 2^{β - α} \leq ε F$ , which is stronger than the claim; hence assume $ε < 2^{β - α}$ and put $h_{0} : = ε^{1 / (β - α)} \in (0, 2)$ . For a pair with $∣ x - y ∣ \geq h_{0}$ use the trivial bound $∣ f (x) - f (y) ∣ \leq 2 M$ , and for a pair with $∣ x - y ∣ < h_{0}$ use the $β$ -Hölder bound: $\frac{∣ f (x) - f (y) ∣}{∣ x - y ∣^{α}} \leq \max {2 M h_{0}^{- α}, F h_{0}^{β - α}} = \max {2 M ε^{- α / (β - α)}, F ε} \leq F ε + 2 M ε^{- α / (β - α)} .$ This is the claim for $R = 1$ with $C_{3} = 2$ .

1.3F2givenalgebra

Interior-point difference estimates, including points near the boundary. Write $M_{j} : = \max_{∣ γ ∣ = j} \sup_{B_{1}} ∣ D^{γ} u ∣$ , $M_{0} = \sup_{B_{1}} ∣ u ∣$ , and $H : = \max_{∣ γ ∣ = k} [D^{γ} u]_{0, α; B_{1}}$ , and set $ρ_{*} : = 1 / (8 n k)$ . For each $x \in B_{1}$ choose an invertible frame $V_{x}$ as follows. If $∣ x ∣ \leq 1 / 2$ , take $V_{x} = I$ . If $∣ x ∣ > 1 / 2$ , put $ν = x / ∣ x ∣$ , choose an orthonormal basis $e_{1}, \dots, e_{n - 1}$ of the tangent space $ν^{⊥}$ , and take the columns of $V_{x}$ to be $e_{i} - ν$ for $1 \leq i < n$ and $- ν$ for the last column (when $n = 1$ there is just the column $- ν$ ). These frames and their inverses have norms bounded by constants depending only on $n$ . Define ${\tilde{u}}_{x} (z) : = u (x + V_{x} z)$ wherever $x + V_{x} z \in B_{1}$ . For any vector $ℓ \in [0, \infty)^{n}$ with $m : = \sum_{i} ℓ_{i} \leq n k$ and any $0 \leq t \leq 1$ , the whole segment $x + t ρ V_{x} ℓ$ lies in $B_{1}$ whenever $0 < ρ \leq ρ_{*}$ . In the inner case, $∣ x + t ρ V_{x} ℓ ∣ \leq 1 / 2 + ρ m \leq 5 / 8$ . In the outer case write $V_{x} ℓ = - m ν + \sum_{i < n} ℓ_{i} e_{i}$ ; its tangential component has norm at most $m$ , so $∣ x + t ρ V_{x} ℓ ∣^{2} \leq r^{2} - 2 t ρ m r + 2 t^{2} ρ^{2} m^{2} \leq r^{2} < 1$ , since $r = ∣ x ∣ \geq 1 / 2$ and $t ρ m \leq 1 / 8 \leq r / 2$ . Thus every sample point and Taylor segment below is contained in $B_{1}$ . For $0 \leq d \leq k - 1$ , choose the unique weights $a_{0}^{(d)}, \dots, a_{k - 1}^{(d)}$ solving the Vandermonde system $\sum_{ℓ = 0}^{k - 1} a_{ℓ}^{(d)} ℓ^{q} = d! 1_{q = d}$ for $q = 0, \dots, k - 1$ . For a multi-index $δ$ of order $j < k$ , define the tensor stencil $Q_{ρ}^{δ} {\tilde{u}}_{x} (0) : = ρ^{- j} \sum_{ℓ \in {0, \dots, k - 1}^{n}} (\prod_{i = 1}^{n} a_{ℓ_{i}}^{(δ_{i})}) {\tilde{u}}_{x} (ρ ℓ) .$ Taylor-expand ${\tilde{u}}_{x}$ at $0$ through degree $k - 1$ . The moment identities make this stencil equal to $D_{z}^{δ} {\tilde{u}}_{x} (0)$ on every polynomial of total degree less than $k$ . The integral remainder at each stencil point is bounded by $C_{n, k} ρ^{k} M_{k}$ using [F2] and the uniform frame bound. The weights and stencil are fixed by $n, k$ , hence $∣ Q_{ρ}^{δ} {\tilde{u}}_{x} (0) - D_{z}^{δ} {\tilde{u}}_{x} (0) ∣ \leq C_{n, k} ρ^{k - j} M_{k}, ∣ Q_{ρ}^{δ} {\tilde{u}}_{x} (0) ∣ \leq C_{n, k} ρ^{- j} M_{0} .$ For a multi-index $δ$ of order $k$ , instead use the ordinary iterated forward difference in the $z$ -coordinates, $Δ_{ρ}^{δ} {\tilde{u}}_{x} (0) : = \sum_{0 \leq γ \leq δ} (- 1)^{k - ∣ γ ∣} (\binom{δ}{γ}) {\tilde{u}}_{x} (ρ γ)$ . Repeated use of the fundamental theorem of calculus gives $ρ^{- k} Δ_{ρ}^{δ} {\tilde{u}}_{x} (0)$ as the average of $D_{z}^{δ} {\tilde{u}}_{x}$ at points $ρ \sum_{q = 1}^{k} t_{q} e_{i_{q}}$ , where the list $i_{1}, \dots, i_{k}$ contains $δ_{i}$ copies of $i$ and $0 \leq t_{q} \leq 1$ . These segments lie in $B_{1}$ by the preceding geometry; the chain rule and the $α$ -Hölder seminorm of the order- $k$ derivatives therefore give $∣ ρ^{- k} Δ_{ρ}^{δ} {\tilde{u}}_{x} (0) - D_{z}^{δ} {\tilde{u}}_{x} (0) ∣ \leq C_{n, k} ρ^{α} H, ∣ Δ_{ρ}^{δ} {\tilde{u}}_{x} (0) ∣ \leq 2^{k} M_{0} .$ Finally, $\partial_{x_{a}} = \sum_{i} (V_{x}^{- 1})_{i a} \partial_{z_{i}}$ , so each canonical derivative of order $j$ is a uniformly bounded linear combination of frame derivatives of that order. We have proved, for every $x \in B_{1}$ , $0 < ρ \leq ρ_{*}$ , and canonical multi-index $β$ , $∣ D^{β} u (x) ∣ \leq C_{n, k} (ρ^{- j} M_{0} + ρ^{k - j} M_{k}) (j = ∣ β ∣ < k), ∣ D^{β} u (x) ∣ \leq C_{n, k} (ρ^{- k} M_{0} + ρ^{α} H) (j = k) .$ Taking suprema gives these same bounds for the full-ball quantities $M_{j}$ ; the estimates are valid up to points arbitrarily close to $\partial B_{1}$ .

2.1step 1.3F1F3casesalgebra

Part (ii). By step 1.3, for every $x \in B_{1}$ , $0 < ρ \leq ρ_{*}$ , and $∣ β ∣ = k$ , $∣ D^{β} u (x) ∣ \leq C_{n, k} (M_{0} ρ^{- k} + ρ^{α} H) .$ If $M_{0} = 0$ , then $u = 0$ . If $H = 0$ and $M_{0} > 0$ , use $ρ = ρ_{*}$ and take the supremum to obtain $M_{k} \leq C (n, k) M_{0}$ . Otherwise assume $M_{0}, H > 0$ and put $s = (M_{0} / H)^{1 / (k + α)}$ . If $s \geq ρ_{*}$ , then $H \leq ρ_{*}^{- (k + α)} M_{0}$ , and the estimate with $ρ = ρ_{*}$ gives $M_{k} \leq C (n, k, α) M_{0}$ . If $s < ρ_{*}$ , take $ρ = s$ to get $M_{k} \leq C_{n, k, α} M_{0}^{α / (k + α)} H^{k / (k + α)}$ . Young's inequality [F3], with $p = (k + α) / k$ and $q = (k + α) / α$ , then gives $M_{k} \leq ε H + C (n, k, α, ε) M_{0}$ . This proves (ii) for $R = 1$ .

3.1step 1.3step 2.1F1algebra

Part (i). By part (ii), for every $δ > 0$ there is $C_{δ}$ such that $M_{k} \leq δ H + C_{δ} M_{0}$ . Choose $δ : = ε / (2 k C_{n, k})$ , increasing $C_{n, k}$ in step 1.3 if necessary so it is at least $1$ . Fix any $ρ \in (0, ρ_{*}]$ . For $0 \leq j < k$ , the lower-order estimate of step 1.3 gives $M_{j} \leq C_{n, k} ρ^{- j} M_{0} + C_{n, k} ρ^{k - j} M_{k} \leq C (n, k, α, ε) M_{0} + \frac{ε}{2 k} H,$ since $ρ_{*} < 1$ . Summing over the $k$ orders yields $\sum_{j = 0}^{k - 1} M_{j} \leq C (n, k, α, ε) M_{0} + \frac{ε}{2} H \leq C M_{0} + ε ∥ u ∥_{k, α; B_{1}}^{*}$ , which is (i) for $R = 1$ .

4.1step 1.1step 1.2step 2.1step 3.1F1given∎

Scaling back and conclusion. Undoing the change of variables of step 1.1 with the scaling identities of [F1] transforms (i), (ii) and (iii) for $R = 1$ into the three displayed statements for general $R$ , with the same constants: $R^{j} M_{j} (u) = M_{j} (v)$ , $R^{k + α} H (u) = H (v)$ , $R^{γ} [f]_{0, γ; B_{R}} = [g]_{0, γ; B_{1}}$ and $\sup_{B_{R}} ∣ f ∣ = \sup_{B_{1}} ∣ g ∣$ . The constants depend only on $n, k, α, ε$ (and on $n, α, β, ε$ in (iii)), never on $u, f, R$ or the centre, and no choice principle is used.

Remarks

Parts (i) and (ii) are the derivative form of the Ehrling inequality: in part (ii), the smallness parameter $ε$ is bought at the price of a constant blowing up like $ε^{- k / α}$ under the displayed Young exponents, which is the price paid in the freezing and Schauder estimates below.
The proof of parts (i) and (ii) uses the top-order Hölder seminorm only through the difference-quotient approximation; no compactness of the embedding $C^{k} ↪ C^{k - 1}$ or Arzelà–Ascoli argument is used, so the estimate is fully quantitative.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Freezing coefficients makes the Schauder error absorbable on a small ball

Statement

Let $n \geq 2$ , $0 < α < 1$ , $R > 0$ and $x_{0} \in R^{n}$ . Let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic on $B_{R} (x_{0})$ with constants $0 < λ \leq Λ < \infty$ , $[A]_{0, α; B_{R} (x_{0})} \leq K$ , and $b, c \in C^{0, α} (B_{R} (x_{0}))$ . Put $L_{0} : = a^{i j} (x_{0}) \partial_{i} \partial_{j}$ and $M_{R} : = R^{α} K + R ∥ b ∥_{\infty} + R^{1 + α} [b]_{0, α} + R^{2} ∥ c ∥_{\infty} + R^{2 + α} [c]_{0, α} .$ Assume $M_{R} < \infty$ . For every $ε > 0$ there are $η_{ε} \in (0, 1]$ and $C_{ε} < \infty$ , depending only on $n, α, λ, Λ, M_{R}, ε$ , such that for every radius $0 < ρ \leq R η_{ε}$ and every $u$ with $∥ u ∥_{2, α; B_{ρ} (x_{0})}^{*} < \infty$ , $ρ^{2} ∥ (L - L_{0}) u ∥_{0, α; B_{ρ} (x_{0})}^{*} \leq ε ∥ u ∥_{2, α; B_{ρ} (x_{0})}^{*} + C_{ε} \sup_{B_{ρ} (x_{0})} ∣ u ∣,$ where the cutoff $R η_{ε}$ is at most $R$ and the constant is uniform over all smaller radii, and $∥ g ∥_{0, α; B_{ρ}}^{*} : = \sup_{B_{ρ}} ∣ g ∣ + ρ^{α} [g]_{0, α; B_{ρ}}$ .

Facts & Assumptions

Given: $n \geq 2$ , $0 < α < 1$ , $R > 0$ , $x_{0}$ , an operator $L$ with the coefficient bounds of the Statement, a fixed $ε > 0$ , and any $0 < ρ \leq R η_{ε}$ with $∥ u ∥_{2, α; B_{ρ} (x_{0})}^{*} < \infty$ .

[F1]

$L_{0}$ is the constant-coefficient operator with matrix $A (x_{0})$ , so $(L - L_{0}) u = (a^{i j} - a^{i j} (x_{0})) \partial_{i} \partial_{j} u + b^{i} \partial_{i} u + c u$ ; the coefficient bounds are recorded by $M_{R}$ in the Statement, and $x_{0}$ is the base point for every frozen coefficient. (Uniformly elliptic nondivergence-form operators and their frozen coefficients, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains)

[F2]

In the normalized variables of step 1.1, put $K : = R^{α} K$ and let $B, C$ be the scaled $C^{0, α}$ bounds of $\tilde{b}, \tilde{c}$ ; then $K + B + C \leq M_{R}$ . On $B_{t}$ , $∣ {\tilde{a}}^{i j} (z) - {\tilde{a}}^{i j} (0) ∣ \leq K t^{α}$ and $[{\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)]_{0, α; B_{t}} \leq K$ . Also $[D_{i} \tilde{u}]_{0, α; B_{t}} \leq (2 t)^{1 - α} \sup_{B_{t}} ∣ D^{2} \tilde{u} ∣$ and $[\tilde{u}]_{0, α; B_{t}} \leq (2 t)^{1 - α} \sup_{B_{t}} ∣ D \tilde{u} ∣$ . (Local Hölder and scaled C-two-alpha norms on balls, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains)

[F3]

Interpolation with $ε$ -loss: for every $ε^{'} > 0$ there is $C_{ε^{'}}$ with (i) $ρ \sup_{B_{ρ}} ∣ D u ∣ \leq ε^{'} ∥ u ∥_{2, α; B_{ρ}}^{*} + C_{ε^{'}} \sup ∣ u ∣$ , (ii) $\sup ∣ D^{2} u ∣ \leq ε^{'} ρ^{α} [D^{2} u]_{0, α; B_{ρ}} + C_{ε^{'}} ρ^{- 2} \sup ∣ u ∣$ , and $ρ^{2 + α} [D^{2} u]_{0, α; B_{ρ}} \leq ∥ u ∥_{2, α; B_{ρ}}^{*}$ . (Ehrling-type Hölder and derivative interpolation with an epsilon loss)

[F4]

The product rule for the Hölder seminorm: $[f g]_{0, α} \leq [f]_{0, α} \sup ∣ g ∣ + \sup ∣ f ∣ [g]_{0, α}$ , and the elementary inequality $a b \leq ε a^{p} + C_{p} ε^{- 1 / (p - 1)} b^{p / (p - 1)}$ for $a, b \geq 0$ and $p > 1$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Young's inequality for conjugate real exponents)

Proof

technique · direct

1.1F1F2F3given

Normalize the scale and record the coefficient oscillation. Put $z = (x - x_{0}) / R$ and $\tilde{u} (z) = u (x_{0} + R z)$ . In these variables the operator has coefficients $\tilde{a} (z) = a (x_{0} + R z)$ , $\tilde{b} (z) = R b (x_{0} + R z)$ and $\tilde{c} (z) = R^{2} c (x_{0} + R z)$ on $B_{1}$ , and their dimensionless Hölder bounds are controlled by $M_{R}$ in the Statement. Write $t : = ρ / R \leq 1$ . Since $x_{0}$ is the centre of the original ball, [F1] gives $\sup_{B_{t}} ∣ {\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0) ∣ \leq (R^{α} K) t^{α}$ and $[{\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)]_{0, α; B_{t}} \leq R^{α} K$ . Fix $ε^{'} > 0$ to be chosen below and let $C_{ε^{'}}$ be the constant of [F3]; all estimates below are in the normalized variables on $B_{t}$ and the scaled norm is $∥ \tilde{u} ∥^{*} : = ∥ \tilde{u} ∥_{2, α; B_{t}}^{*}$ .

2.1step 1.1F2F3F4algebra

The second-order part. By [F4] and step 1.1, $[({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u}]_{0, α} \leq K \sup ∣ D^{2} \tilde{u} ∣ + K t^{α} [D^{2} \tilde{u}]_{0, α}$ and $\sup ∣ ({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u} ∣ \leq K t^{α} \sup ∣ D^{2} \tilde{u} ∣$ . Hence F3 and its last bound give $t^{2} (\sup ∣ ({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u} ∣ + t^{α} [({\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0)) \partial_{i} \partial_{j} \tilde{u}]_{0, α}) \leq [ε^{'} (K + K t^{α}) + C K t^{α}] ∥ \tilde{u} ∥^{*} + C_{ε^{'}} K t^{α} \sup ∣ \tilde{u} ∣ .$ The $C K t^{α} ∥ \tilde{u} ∥^{*}$ contribution is the product-seminorm term $\sup ∣ {\tilde{a}}^{i j} - {\tilde{a}}^{i j} (0) ∣ [D_{i} D_{j} \tilde{u}]_{0, α}$ after scaling; it has no interpolation factor $ε^{'}$ .

2.2step 1.1F2F3F4algebra

The lower-order part. Write $\tilde{M} : = ∥ \tilde{b} ∥_{C^{0, α}} + ∥ \tilde{c} ∥_{C^{0, α}} \leq M_{R}$ . By [F4] and [F2], the supremum of ${\tilde{b}}^{i} \partial_{i} \tilde{u} + \tilde{c} \tilde{u}$ is bounded by $\tilde{M} (\sup ∣ D \tilde{u} ∣ + \sup ∣ \tilde{u} ∣)$ , and its Hölder seminorm is bounded by $\tilde{M} (\sup ∣ D \tilde{u} ∣ + (2 t)^{1 - α} \sup ∣ D^{2} \tilde{u} ∣ + \sup ∣ \tilde{u} ∣ + (2 t)^{1 - α} \sup ∣ D \tilde{u} ∣)$ . Multiplying by $t^{2}$ and $t^{2 + α}$ , respectively, and inserting F3,(ii) shows that this contribution is at most $ε^{'} C_{1}^{l o w} (t) ∥ \tilde{u} ∥^{*} + C_{n, α, ε^{'}} (1 + \tilde{M}) \sup ∣ \tilde{u} ∣$ , where $C_{1}^{l o w} (t)$ is bounded for $0 < t \leq 1$ and has a finite limit as $t ↓ 0$ .

3.1step 1.1step 2.1step 2.2F3givenalgebra∎

Uniform choice of the normalized radius and conclusion. In normalized variables, collect the top-norm coefficients from steps 2.1 and 2.2 as $C_{e r r} (t) : = ε^{'} C_{i n t e r p} (t) + C_{o s c} K t^{α}$ , where $C_{i n t e r p} (t)$ is bounded on $0 < t \leq 1$ and has a finite limit at $0$ , and $C_{o s c}$ depends only on $n, α, λ, Λ$ . The second term explicitly includes the product-seminorm term of step 2.1, which has no $ε^{'}$ factor and tends to zero as $t ↓ 0$ . Given $ε > 0$ , choose first $ε^{'} > 0$ so that $ε^{'} C_{i n t e r p} (0) \leq ε / 4$ (if $C_{i n t e r p} (0) = 0$ , any positive $ε^{'}$ suffices); then choose $η_{ε} \in (0, 1]$ so small that $ε^{'} C_{i n t e r p} (t) \leq ε / 2$ and $C_{o s c} K t^{α} \leq ε / 2$ for every $0 < t \leq η_{ε}$ . Thus $C_{e r r} (t) \leq ε$ uniformly over every such radius. The lower-order remainder coefficients are also uniformly bounded there by $C_{ε}$ depending only on the displayed dimensionless parameters; the cap $t \leq 1$ is the small-scale condition used for those terms. Scaling back gives the same estimate for every physical radius $0 < ρ \leq R η_{ε}$ .

Remarks

The quantitative structure is the classical one: after normalization, the oscillation of the principal coefficients on a radius- $t$ ball is at most $(R^{α} K) t^{α}$ ; the frozen error carries the two extra derivatives scaled as $t^{2}$ , and interpolation converts the resulting powers into an arbitrarily small multiple of the full scaled norm plus a bounded multiple of $\sup ∣ u ∣$ .
The estimate is uniform over every smaller radius below $R η_{ε}$ . The cutoff fraction $η_{ε} \leq 1$ is chosen from the dimensionless coefficient bounds, including the small-scale cap needed for the lower-order terms.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Interior Schauder estimate for uniformly elliptic equations

Statement

Assume Countable Choice. Let $n \geq 2$ , $0 < α < 1$ , $R > 0$ , $x_{0} \in R^{n}$ , and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic on $B_{R} (x_{0})$ with constants $λ, Λ$ , $[A]_{0, α; B_{R} (x_{0})} \leq K$ , and $b, c \in C^{0, α} (B_{R} (x_{0}))$ with finite global Hölder seminorms. Let $u \in C^{2} (B_{R} (x_{0})) \cap L^{\infty} (B_{R} (x_{0}))$ satisfy $L u = f$ pointwise with $f \in C^{0, α} (B_{R} (x_{0}))$ . Then $u \in C^{2, α} (B_{R / 2} (x_{0}))$ and $∥ u ∥_{2, α; B_{R / 2} (x_{0})}^{*} \leq C (∥ u ∥_{\infty; B_{R} (x_{0})} + R^{2} ∥ f ∥_{\infty; B_{R} (x_{0})} + R^{2 + α} [f]_{0, α; B_{R} (x_{0})}),$ where $C$ depends only on $n, α, λ, Λ$ and the finite dimensionless coefficient bounds $R^{α} K + R ∥ b ∥_{\infty} + R^{1 + α} [b]_{0, α} + R^{2} ∥ c ∥_{\infty} + R^{2 + α} [c]_{0, α}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 2$ , $0 < α < 1$ , $R > 0$ , $x_{0}$ , the operator $L$ with the stated bounds, $u \in C^{2} (B_{R} (x_{0})) \cap L^{\infty}$ and $f \in C^{0, α} (B_{R} (x_{0}))$ with $L u = f$ pointwise.

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ , used through the measure, potential and estimate interfaces below. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The scaled norms are $∥ w ∥_{2, α; B_{ρ}}^{*} = \sum_{j = 0}^{2} ρ^{j} \max_{∣ β ∣ = j} \sup_{B_{ρ}} ∣ D^{β} w ∣ + ρ^{2 + α} \max_{∣ β ∣ = 2} [D^{β} w]_{0, α; B_{ρ}}$ and $∥ g ∥_{0, α; B_{ρ}}^{*} = \sup_{B_{ρ}} ∣ g ∣ + ρ^{α} [g]_{0, α; B_{ρ}}$ ; the scaling identity $∥ w \circ σ_{ρ} ∥_{2, α; B_{1}}^{*} = ∥ w ∥_{2, α; B_{ρ}}^{*}$ holds for $σ_{ρ} (z) = x_{0} + ρ z$ . (Local Hölder and scaled C-two-alpha norms on balls, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains)

[F2]

The interior Poisson (Laplace) estimate: for $n \geq 2$ , $0 < α < 1$ and $v \in C^{2} (B_{τ}) \cap L^{\infty} (B_{τ})$ with $- Δ v = g \in C^{0, α} (B_{τ})$ pointwise, one has $∥ v ∥_{2, α; B_{τ / 2}}^{*} \leq C_{n, α} (∥ v ∥_{\infty; B_{τ}} + τ^{2} ∥ g ∥_{\infty; B_{τ}} + τ^{2 + α} [g]_{0, α; B_{τ}})$ . (Interior estimate for the Poisson equation with Hölder data)

[F3]

Uniform small-ball freezing (Freezing coefficients makes the Schauder error absorbable on a small ball): for every $ϵ > 0$ there are $η_{ϵ} \in (0, 1]$ and $C_{ϵ}$ , depending only on the dimensionless coefficient bounds and ellipticity, such that the freezing bound holds for every $0 < r \leq R η_{ϵ}$ on a ball centered at the frozen point. For a recentered patch with center $x \in B_{3 R / 4} (x_{0})$ , apply that lemma on the ambient ball $B_{R / 8} (x)$ , which lies in $B_{7 R / 8} (x_{0})$ and has dimensionless coefficient bounds no larger than the original ones; hence the same normalized cutoff gives the bound for every $0 < r \leq R η_{ϵ} / 8$ on that patch.

[F4]

Linear normalization and ellipsoid-to-ball geometry. If $A_{0}$ is symmetric positive definite with spectrum in $[λ, Λ]$ , put $S = A_{0}^{1 / 2}$ and $v (y) = w (x + S y)$ . Then $A_{0} : D_{x}^{2} w (x + S y) = Δ_{y} v (y)$ , $∥ S ∥ \leq \sqrt{Λ}$ and $∥ S^{- 1} ∥ \leq 1 / \sqrt{λ}$ (Uniformly elliptic nondivergence-form operators and their frozen coefficients, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). For every $r > 0$ , $S (B_{r / \sqrt{Λ}}) \subseteq B_{r}$ and $B_{θ_{g e o} r} \subseteq S (B_{r / (2 \sqrt{Λ})})$ with $θ_{g e o} : = \frac{1}{2} \sqrt{λ / Λ}$ . Consequently, for every $0 < θ \leq θ_{g e o}$ , the Poisson estimate on $B_{r / \sqrt{Λ}}$ controls the original solution on $B_{θ r}$ , with norm-comparison constants depending only on $λ, Λ$ .

[F5]

Local $L^{2}$ Hessian bound for Newtonian potentials (Local $W^{2, p}$ regularity of weak solutions of the Poisson equation): if $g \in L_{c}^{2} (R^{n})$ and $w = N g$ , then $w \in W_{l o c}^{2, 2}$ and its Hessian has the local estimate proved by Newtonian–Riesz representation. For $g$ supported in a ball $B_{r}$ with $\int g = 0$ , rescaling the fundamental solution and using the local Young bound for $N$ and $\nabla N$ gives, for every fixed $C_{0} > 1$ , $∥ w ∥_{L^{2} (B_{C_{0} r})} \leq C r^{2} ∥ g ∥_{L^{2} (B_{r})}$ ; in dimension $2$ the logarithmic scaling term vanishes because $\int g = 0$ . Rescaling the local $W^{2, 2}$ estimate on concentric balls then gives $∥ D^{2} w ∥_{L^{2} (B_{r})} \leq C ∥ g ∥_{L^{2} (B_{r})}$ . The same conclusion for a frozen operator $A_{0} : D^{2}$ follows from [F4], with constants depending on $n, λ, Λ$ . (Local $W^{2, p}$ regularity of weak solutions of the Poisson equation)

[F6]

Harmonic excess decay. If $h$ is $A_{0}$ -harmonic on $B_{r} (x)$ , then for every sufficiently small $θ = θ (n, λ, Λ) \in (0, θ_{g e o}]$ , ${(\frac{1}{∣ B_{θ r} (x) ∣} \int_{B_{θ r} (x)} ∣ D^{2} h - (D^{2} h)_{B_{θ r} (x)} ∣^{2})}^{1 / 2} \leq C_{0} θ {(\frac{1}{∣ B_{r} (x) ∣} \int_{B_{r} (x)} ∣ D^{2} h - (D^{2} h)_{B_{r} (x)} ∣^{2})}^{1 / 2} .$ Indeed each component of $D^{2} h - (D^{2} h)_{B_{r}}$ is harmonic; transform by [F4], apply the interior derivative estimate for harmonic functions, and compare the contained and containing balls. Weakly harmonic components are smooth by Locally integrable weakly harmonic functions are smooth, and the derivative bound is Interior derivative estimates for harmonic functions.

[F7]

For $G \in L^{1} (B)$ , write $(G)_{B} : = ∣ B ∣^{- 1} \int_{B} G$ ; this is the ball-average operator of The average of a locally integrable function over a Euclidean ball. In particular, $(G)_{B_{r} (x)}$ is constant in $x$ for each fixed ball and the volume ratio of concentric balls is $(r / s)^{n}$ .

Proof

technique · direct

1.1F1F2F4givenalgebra

Ellipsoid geometry for freezing. Fix $x \in B_{3 R / 4} (x_{0})$ , a radius $0 < r \leq R / 8$ with $B_{r} (x) \subset B_{7 R / 8} (x_{0})$ , and set $A_{0} = A (x)$ , $S = A_{0}^{1 / 2}$ , and $τ = r / \sqrt{Λ}$ . For $v (y) : = u (x + S y)$ , [F4] gives $Δ v = (L_{x} u) \circ (x + S \cdot)$ on $B_{τ}$ , since $S (B_{τ}) \subset B_{r} (x)$ . Equivalently, $- Δ v = - f \circ (x + S \cdot) + ((L - L_{x}) u) \circ (x + S \cdot)$ , so the estimate [F2] applies with this signed right-hand side. Fix any $0 < θ \leq θ_{g e o}$ ; the target ball $B_{θ r} (x)$ lies in $x + S (B_{τ / 2})$ . Thus [F2], restricted from this ellipsoid to the target ball, yields $∥ u ∥_{2, α; B_{θ r} (x)}^{*} \leq C (∥ u ∥_{\infty; B_{r} (x)} + r^{2} ∥ f ∥_{\infty; B_{r} (x)} + r^{2 + α} [f]_{0, α; B_{r} (x)} + r^{2} ∥ (L - L_{x}) u ∥_{0, α; B_{r} (x)}^{*}),$ with $C = C (n, α, λ, Λ)$ . This estimate is conditional on the Hessian having a finite Hölder seminorm on the patch. The independent bootstrap below establishes that condition before its later quantitative use; $C^{2}$ alone does not establish it.

1.2F1F7givenalgebra

A noncircular Campanato bootstrap. Put $H = D^{2} u$ and $Q : = 1 + ∥ u ∥_{C^{2} (\overline{B_{7 R / 8} (x_{0})})} + ∥ f ∥_{C^{0, α} (B_{7 R / 8})} + ∥ b ∥_{C^{0, α} (B_{R})} + ∥ c ∥_{C^{0, α} (B_{R})}$ . For $x \in B_{3 R / 4} (x_{0})$ and $0 < r \leq R / 8$ , let $E_{x} (r) : = {(\frac{1}{∣ B_{r} (x) ∣} \int_{B_{r} (x)} ∣ H - (H)_{B_{r} (x)} ∣^{2})}^{1 / 2}, g_{x} : = f - b \cdot D u - c u - (A - A (x)) : D^{2} u .$ Then $A (x) : D^{2} u = g_{x}$ pointwise. The product oscillation estimate, $[A]_{0, α} \leq K$ , and the boundedness of $D u, D^{2} u$ on $\overline{B_{7 R / 8}}$ give ${osc}_{2} (g_{x}; B_{r} (x)) \leq C K r^{α} (E_{x} (r) + ∣ (H)_{B_{r} (x)} ∣) + C_{Q} r^{α} \leq C r^{α} E_{x} (r) + C_{Q} r^{α},$ where ${osc}_{2} (q; B) : = (\frac{1}{∣ B ∣} \int_{B} ∣ q - (q)_{B} ∣^{2})^{1 / 2}$ ; $C_{Q}$ is finite and may depend on the preliminary bound $Q$ . No Hölder regularity of $D^{2} u$ is used here.

2.1step 1.2F4F5F6F7A1algebra

Constant-coefficient replacement and excess decay. Let $m_{x} = (g_{x})_{B_{r} (x)}$ . Extend $g_{x} - m_{x}$ by zero from $B_{r} (x)$ , transform the frozen operator by [F4], and take $w_{x}$ to be the negative of its Newtonian potential, so $A (x) : D^{2} w_{x} = g_{x} - m_{x}$ because $N$ is normalized by $- Δ N g = g$ . The datum has mean zero, so [F5] gives $∥ D^{2} w_{x} ∥_{L^{2} (B_{r} (x))} \leq C ∥ g_{x} - m_{x} ∥_{L^{2} (B_{r} (x))}$ . Choose a quadratic $q_{x}$ with $A (x) : D^{2} q_{x} = m_{x}$ and put $h_{x} = u - w_{x} - q_{x}$ . Then $h_{x}$ is weakly $A (x)$ -harmonic on $B_{r} (x)$ ; its Hessian is smooth there by [F6], and $D^{2} q_{x}$ is constant. Apply [F6] to $h_{x}$ , use $E_{x} (r) \leq E_{h_{x}} (r) + C {osc}_{2} (g_{x}; B_{r})$ and the volume ratio between $B_{r}$ and $B_{θ r}$ to obtain $E_{x} (θ r) \leq C_{0} θ E_{x} (r) + C_{1} θ^{- n / 2} {osc}_{2} (g_{x}; B_{r}) \leq q E_{x} (r) + C_{Q} r^{α} .$ Choose $θ \leq θ_{g e o}$ so $C_{0} θ < \frac{1}{4} θ^{α}$ , then choose a uniform $r_{0} \leq R / 8$ so $C_{1} θ^{- n / 2} C K r_{0}^{α} < \frac{1}{4} θ^{α}$ ; thus $q < \frac{1}{2} θ^{α}$ . Iterating over $r_{j} = θ^{j} r_{0}$ and using $E_{x} (r_{0}) \leq 2 ∥ H ∥_{L^{\infty} (B_{7 R / 8})}$ gives $E_{x} (r_{j}) \leq C_{Q} r_{j}^{α}$ . For intermediate radii, $E_{x} (s) \leq (r_{j} / s)^{n / 2} E_{x} (r_{j})$ when $r_{j + 1} < s \leq r_{j}$ , so the same bound holds for every $0 < s \leq r_{0}$ , uniformly for $x \in B_{3 R / 4}$ .

3.1step 2.1F7givenalgebra

Campanato embedding on the nested-patch region. Since $H$ is continuous, the means $(H)_{B_{s} (x)}$ converge to $H (x)$ as $s ↓ 0$ . Telescoping the dyadic means and using step 2.1 gives $∣ H (x) - (H)_{B_{s} (x)} ∣ \leq C_{Q} s^{α}$ for every $x \in B_{3 R / 4}$ and $0 < s \leq r_{0}$ . If $x, y \in B_{5 R / 8}$ and $d : = ∣ x - y ∣ < r_{0} / 8$ , then $B_{2 d} (x)$ and $B_{2 d} (y)$ are both contained in $B_{4 d} (x)$ ; comparing each of their means with $(H)_{B_{4 d} (x)}$ costs only the fixed volume ratio $∣ B_{4 d} ∣ / ∣ B_{2 d} ∣ = 2^{n}$ and is bounded by $C E_{x} (4 d)$ . Since $4 d < r_{0} / 2$ , the Step 2.1 excess estimate applies at this radius, and the telescoping estimates for $x$ and $y$ give $∣ H (x) - H (y) ∣ \leq C_{Q} d^{α}$ . For $d \geq r_{0} / 8$ , the preliminary bound $2 ∥ H ∥_{\infty} \leq C_{Q} d^{α}$ applies. Therefore $D^{2} u \in C^{0, α} (B_{5 R / 8})$ , in particular on $B_{R / 2}$ , before the quantitative Schauder estimate is invoked. The preliminary constant $C_{Q}$ is used only to prove finiteness.

4.1F2F3step 1.1algebrastep 3.1

A local finite-norm estimate with corrected nested radii. Now that the norm is finite, apply [F3] in step 1.1 with arbitrary small error $ϵ$ to the term $(L - L_{x}) u$ on any recentered patch of radius $r \leq R η_{ϵ} / 8$ . Use part (i) of the Hölder interpolation lemma to bound the first-derivative term in the local scaled norm by an arbitrarily small multiple of $r^{2} ∥ D^{2} u ∥_{\infty} + r^{2 + α} [D^{2} u]_{0, α}$ plus $C_{ϵ} ∥ u ∥_{\infty}$ . Thus, for any prescribed $η > 0$ , the choices of the freezing and interpolation parameters give $∥ u ∥_{2, α; B_{ϑ r} (x)}^{*} \leq C_{η} (∥ u ∥_{\infty; B_{r} (x)} + r^{2} ∥ f ∥_{\infty; B_{r} (x)} + r^{2 + α} [f]_{0, α; B_{r} (x)}) + η (r^{2} ∥ D^{2} u ∥_{\infty; B_{r} (x)} + r^{2 + α} [D^{2} u]_{0, α; B_{r} (x)}) .$ The constant $C_{η}$ may depend on the dimensionless coefficient bounds, ellipticity, and the chosen normalized patch radius; the small factor $η$ multiplies only the top-order terms after interpolation.

5.1step 3.1step 4.1F1F3inductionalgebra

Hole filling with interpolation of the separated-pair term. Write $H (s) : = R^{2 + α} \max_{∣ β ∣ = 2} [D^{β} u]_{0, α; B_{s} (x_{0})}$ , $M (s) : = R^{2} \max_{∣ β ∣ = 2} \sup_{B_{s} (x_{0})} ∣ D^{β} u ∣$ , and $S_{R} : = ∥ u ∥_{\infty; B_{R}} + R^{2} ∥ f ∥_{\infty; B_{R}} + R^{2 + α} [f]_{0, α; B_{R}}$ . Fix $γ \in (0, 2^{- (2 + α)})$ . For $R / 2 \leq s < t \leq 5 R / 8$ , put $r = (t - s) / 2$ , requiring $r \leq R η_{ϵ} / 8$ as in step 4.1. Apply that estimate centered at every $x \in B_{s} (x_{0})$ , with one fixed target fraction $ϑ > 0$ . If $x, y \in B_{s}$ and $∣ x - y ∣ < ϑ r / 2$ , the pair lies in the target ball centered at $x$ , so dividing its local Hessian seminorm estimate by $r^{2 + α}$ bounds the corresponding quotient. For pairs with $∣ x - y ∣ \geq ϑ r / 2$ , use $∣ D^{β} u (x) - D^{β} u (y) ∣ \leq 2 \sup_{B_{t}} ∣ D^{β} u ∣$ . Thus, for $0 < η \leq 1$ , $H (s) \leq C_{η} (R / r)^{2 + α} S_{R} + C_{0} η H (t) + C_{1} (R / r)^{α} M (t),$ with $C_{0}, C_{1}$ fixed by $n, α, ϑ$ ; the last term includes both the local Hessian-sup error and the separated pairs. Part (ii) of the interpolation lemma on $B_{t}$ , with $R / 2 \leq t \leq 5 R / 8$ , gives $M (t) \leq ζ H (t) + C (1 + ζ^{- 2 / α}) ∥ u ∥_{\infty; B_{t}}$ for $0 < ζ \leq 1$ : this quantitative dependence follows directly from its order-two difference estimate by choosing the normalized difference scale proportional to $ζ^{1 / α}$ . Choose $η$ with $C_{0} η \leq γ / 2$ , then choose $ζ = \min {1, γ / (2 C_{1})} (r / R)^{α}$ . The term containing $M (t)$ is at most $γ H (t) / 2 + C_{γ} (R / r)^{2 + α} S_{R}$ . Consequently $H (s) \leq C_{γ} (R / r)^{2 + α} S_{R} + γ H (t) .$ After these choices, let $δ_{0} = \min {R / 16, R η_{ϵ} / 4}$ , $δ_{j} = δ_{0} 2^{- j}$ , $s_{0} = R / 2$ and $s_{j + 1} = s_{j} + δ_{j}$ . Then $r_{j} = δ_{j} / 2$ meets the local-radius restriction and $s_{j} ↑ s_{\infty} \leq 5 R / 8$ . Iteration has data series bounded by $C_{γ} (2 R / δ_{0})^{2 + α} \sum_{j \geq 0} (γ 2^{2 + α})^{j} S_{R}$ ; its terminal term tends to zero because $H (s_{j}) \leq H (5 R / 8) < \infty$ by step 3.1. Hence $H (R / 2) \leq C S_{R}$ . Parts (ii) and (i) of the interpolation lemma, now at a fixed parameter on $B_{R / 2}$ , bound the scaled second- and first-derivative suprema by $C (H (R / 2) + ∥ u ∥_{\infty})$ . This proves the full scaled norm estimate, with constants depending only on the stated dimensionless bounds and not on the preliminary bootstrap constant $Q$ .

6.1step 3.1step 5.1given∎

Conclusion. Step 3.1 establishes the claimed local regularity noncircularly, and step 5.1 proves the displayed quantitative estimate on the original half ball. The constant is uniform in $R$ when the stated dimensionless coefficient bounds are uniform. The strict range $0 < α < 1$ enters through the Poisson estimate, the freezing lemma, and the Campanato iteration; no boundary condition is used.

Remarks

The two ingredients are exactly the constant-coefficient estimate for the frozen operator and the small-ball absorption of the coefficient oscillation; the lower-order coefficients are treated as data inside the freezing error.
The dependence of the constant on $R$ enters only through the ratio $R / ρ$ , where $ρ$ is determined by the dimensionless coefficient bounds; this is why those bounds are the natural parameters of the estimate.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

$C^{2, α}$ boundary flattening preserves the nondivergence structure, ellipticity and Hölder norms

Statement

Let $n \geq 2$ , $0 < α < 1$ and let $Ω$ be a bounded $C^{2, α}$ domain. For every $x_{0} \in \partial Ω$ , after a rigid motion and possibly reversing the last coordinate, there are $r > 0$ and $φ \in C^{2, α} (Q_{r}^{'})$ , $Q_{r}^{'} = (- r, r)^{n - 1}$ , with $φ (0) = 0$ , $D φ (0) = 0$ , such that for $Q_{r} = Q_{r}^{'} \times (- r, r)$ the shear $Ψ (y^{'}, y_{n}) = (y^{'}, y_{n} + φ (y^{'}))$ is a diffeomorphism onto the patch $U : = Ψ (Q_{r})$ and $Ψ (Q_{r}^{+}) = U \cap Ω, Ψ (Q_{r}^{'} \times {0}) = U \cap \partial Ω,$ where $Q_{r}^{+} = Q_{r}^{'} \times (0, r)$ . In particular the chart is stated on its actual image patch; no equality with the intersection of a Euclidean ball and $Ω$ is asserted. Composition with $Ψ$ gives equivalent $C^{2, α}$ norms on the closures of $U \cap Ω$ and $Q_{r}^{+}$ , with constants depending on the chart. If $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ has $C^{0, α}$ coefficients on $U$ , then its pullback under $v = u \circ Ψ$ is again nondivergence form with $C^{0, α}$ coefficients. Its principal matrix is $\tilde{A} (y) = D Ψ (y)^{- 1} A (Ψ (y)) D Ψ (y)^{- T},$ so its ellipticity constants may be taken as $λ ∥ D Ψ ∥_{\infty}^{- 2}$ and $Λ ∥ D Ψ^{- 1} ∥_{\infty}^{2}$ ; the lower-order coefficients are given by the chain rule and have Hölder bounds controlled by the chart and original coefficient norms.

Facts & Assumptions

Given: $n \geq 2$ , $0 < α < 1$ , a bounded $C^{2, α}$ domain $Ω$ in the graph sense, a boundary point $x_{0} \in \partial Ω$ , and a uniformly elliptic operator $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ with $C^{0, α}$ coefficients on a neighbourhood of $x_{0}$ .

[F1]

There are a rigid motion $R (p) = Q p + b$ , a ball $B \subseteq R^{n - 1}$ and $h \in C^{2, α} (B)$ with $R (Ω \cap W) = R (W) \cap {s < h (y)}$ for a neighbourhood $W$ of $x_{0}$ . For $F (y, s) = s - h (y)$ the gradient $(- D h, 1)$ is nonzero. The graph theorem A regular level set is locally a $C^{k}$ graph of dimension $m - n$ reparametrizes its zero set over the tangent hyperplane; its derivative formula, followed by one differentiation, expresses the new first and second derivatives using those of $h$ and the inverse of a nonvanishing normal derivative. On a smaller compact patch that denominator is bounded away from zero. Products, inversion of the scalar denominator, and Lipschitz composition preserve the $α$ -Hölder bound of $D^{2} h$ , so the new graph is $C^{2, α}$ . After translating and rotating the coordinates (a rigid motion), we may assume $x_{0} = 0$ , the graph passes through the origin and is tangent to ${s = 0}$ there; the one-sided subgraph convention and the regularity class are unchanged. (Bounded C^k domains and boundary charts, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains)

[F2]

A shear $Ψ (y^{'}, y_{n}) = (y^{'}, y_{n} + φ (y^{'}))$ with $φ \in C^{2, α}$ satisfies $J : = D Ψ = (\begin{array}{cc} I & 0 \\ D φ & 1 \end{array})$ , $\det J = 1$ , and $J^{- 1} = D Ψ^{- 1} = (\begin{array}{cc} I & 0 \\ - D φ & 1 \end{array})$ ; it is a $C^{2, α}$ diffeomorphism onto its image and its inverse has the same shear form with $- φ$ . ( $C^{k}$ maps and multi-index derivative notation in Euclidean space, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ )

[F3]

The chain rule gives, for $v = u \circ Ψ$ , $\partial_{j} v (y) = \sum_{k} \partial_{k} u (Ψ (y)) \partial_{j} Ψ_{k} (y)$ and $\partial_{i} \partial_{j} v (y) = \sum_{k, l} \partial_{k} \partial_{l} u (Ψ (y)) \partial_{i} Ψ_{k} \partial_{j} Ψ_{l} + \sum_{k} \partial_{k} u (Ψ (y)) \partial_{i} \partial_{j} Ψ_{k}$ ; for bounded $α$ -Hölder factors, subtracting the product values gives $[f g]_{0, α} \leq ∥ f ∥_{\infty} [g]_{0, α} + ∥ g ∥_{\infty} [f]_{0, α}$ . Composition with a Lipschitz inner map $G$ gives $[f \circ G]_{0, α} \leq [f]_{0, α} Lip (G)^{α}$ , directly from $∣ G (x) - G (y) ∣ \leq Lip (G) ∣ x - y ∣$ . Boundedness is preserved by composition and products. The shears here and their inverses are Lipschitz on their patches: bounded $D φ$ controls the difference of $φ$ at any two points of the convex base box. (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ )

[F4]

$L$ is uniformly elliptic with constants $λ \leq Λ$ , that is $λ ∣ ξ ∣^{2} \leq a^{i j} (x) ξ_{i} ξ_{j} \leq Λ ∣ ξ ∣^{2}$ for all $ξ$ and almost every $x$ ; for a real matrix $T$ , $∣ T η ∣ \leq ∥ T ∥ ∣ η ∣$ and $∣ T η ∣ \geq ∥ T^{- 1} ∥^{- 1} ∣ η ∣$ whenever $T$ is invertible. (Uniformly elliptic nondivergence-form operators and their frozen coefficients)

Proof

technique · direct

1.1F1algebra

Straightening the boundary. By [F1] there is a rigid motion carrying $x_{0}$ to $0$ and the boundary near $x_{0}$ to a graph $s = h (y^{'})$ over a ball $B$ , with $Ω$ on the side $s < h (y^{'})$ , $h (0) = 0$ and $D h (0) = 0$ ; choose $r > 0$ small enough that $\overline{Q_{r}^{'}} \subset B$ and $Ψ (\overline{Q_{r}}) \subset W$ . All derivatives of the graph are then bounded and have the stated Hölder bounds on this smaller patch. Reverse the last coordinate, $z_{n} = - s$ ; then $Ω$ is locally ${z_{n} > - h (y^{'})}$ , and with $φ : = - h$ , $φ (0) = 0$ , $D φ (0) = 0$ and $φ \in C^{2, α} (B_{r})$ : the domain is locally the region above the graph of $φ$ .

2.1step 1.1F1F2algebra

The shear is the chart. Let $Ψ (y^{'}, y_{n}) = (y^{'}, y_{n} + φ (y^{'}))$ ; by [F2] it is a $C^{2, α}$ diffeomorphism with $D Ψ = (\begin{matrix} I & 0 \\ D φ & 1 \end{matrix})$ , $\det D Ψ = 1$ , $D Ψ^{- 1} = (\begin{matrix} I & 0 \\ - D φ & 1 \end{matrix})$ . For $(y^{'}, y_{n}) \in Q_{r}^{+}$ the image point has last coordinate $y_{n} + φ (y^{'}) > φ (y^{'})$ , hence lies above the graph and therefore in $Ω$ ; conversely, if a point $(y^{'}, s)$ of the chart lies in $Ω$ , then $s > φ (y^{'})$ , so $y_{n} : = s - φ (y^{'}) \in (0, r)$ for $s$ in the chart box, and $(y^{'}, s) = Ψ (y^{'}, y_{n})$ . Hence $Ψ (Q_{r}^{+}) = U \cap Ω$ , and $y_{n} = 0$ gives exactly the graph points, so $Ψ (Q_{r}^{'} \times {0}) = U \cap \partial Ω$ .

3.1step 2.1F2F3algebra

Norm equivalence. By [F3], the chain rule expresses each derivative of $v = u \circ Ψ$ of order at most two as a finite sum of products of derivatives of $u \circ Ψ$ and derivatives of $Ψ$ . For the top-order seminorm, $[D^{2} u \circ Ψ]_{0, α; Q_{r}^{+}} \leq [D^{2} u]_{0, α; U \cap Ω} Lip (Ψ)^{α}$ , while the lower-order factor obeys $[D u \circ Ψ]_{0, α; Q_{r}^{+}} \leq diam (Q_{r}^{+})^{1 - α} ∥ D_{y} (D u \circ Ψ) ∥_{L^{\infty} (Q_{r}^{+})} \leq C_{Ψ} ∥ D^{2} u ∥_{L^{\infty} (U \cap Ω)}$ ; the corresponding bound for $u \circ Ψ$ follows from $∥ D u ∥_{\infty}$ . The derivatives $D Ψ$ are Lipschitz with constants controlled by $∥ D^{2} Ψ ∥_{\infty}$ , and $D^{2} Ψ$ is $C^{0, α}$ , so the product seminorms are bounded by $C_{Ψ} ∥ u ∥_{C^{2, α} (U \cap Ω)}$ . Applying the same estimates to $Ψ^{- 1}$ gives the reverse norm inequality. Thus the $C^{2, α}$ norms on $U \cap Ω$ and $Q_{r}^{+}$ are equivalent, with constants depending only on the chart.

3.2step 2.1F2F3algebra

Pullback of the operator. Let $J : = D Ψ$ , $u$ be $C^{2}$ on $U \cap Ω$ , and $v = u \circ Ψ$ . The chain rule gives $D_{y} v = J^{T} D_{x} u$ and $D_{y}^{2} v = J^{T} (D_{x}^{2} u) J + \sum_{k} u_{x_{k}} (Ψ (y)) D_{y}^{2} Ψ_{k} (y) .$ Thus $D_{x} u = J^{- T} D_{y} v$ and $D_{x}^{2} u = J^{- T} ⁣ (D_{y}^{2} v - \sum_{k} (J^{- T} D_{y} v)_{k} D_{y}^{2} Ψ_{k}) J^{- 1} .$ Writing $A (y) : = (a^{i j} (Ψ (y)))$ , substitution into $L u (Ψ (y))$ yields the transformed principal matrix $\tilde{A} = J^{- 1} A J^{- T}$ . More explicitly, the coefficient of $\partial_{y_{m}} v$ is ${\tilde{b}}^{m} = \sum_{i} b^{i} (Ψ) (J^{- 1})_{m i} - \sum_{a, b, k} {\tilde{a}}^{a b} (J^{- T})_{k m} \partial_{y_{a} y_{b}} Ψ_{k},$ and the zero-order coefficient is $\tilde{c} = c \circ Ψ$ ; the minus sign is the one from solving the Hessian identity for $D_{x}^{2} u$ . By [F3] these coefficients are $C^{0, α}$ on the compact patch with Hölder norms bounded in terms of the chart and the original coefficient norms, since $J^{- 1}$ is $C^{1, α}$ and $D^{2} Ψ$ is $C^{0, α}$ .

4.1step 3.2F2F4algebra∎

Ellipticity. For $ξ \in R^{n}$ , set $ζ = J^{- T} ξ$ . Then $\tilde{A} ξ \cdot ξ = A (Ψ) ζ \cdot ζ$ . Since $∣ ζ ∣ \geq ∥ J ∥_{\infty}^{- 1} ∣ ξ ∣$ and $∣ ζ ∣ \leq ∥ J^{- 1} ∥_{\infty} ∣ ξ ∣$ , uniform ellipticity of $A$ gives $λ ∥ J ∥_{\infty}^{- 2} ∣ ξ ∣^{2} \leq \tilde{A} ξ \cdot ξ \leq Λ ∥ J^{- 1} ∥_{\infty}^{2} ∣ ξ ∣^{2} .$ The matrix $\tilde{A} = J^{- 1} A J^{- T}$ is symmetric because $A$ is symmetric. Hence the pullback is uniformly elliptic and the chart maps the boundary problem on $U \cap Ω$ to a half-box problem on $Q_{r}^{+}$ without changing the nondivergence structure.

Remarks

The determinant of the shear is one, so the chart is volume preserving; the metric distortion is entirely in the coefficient transformation $\tilde{A}$ and in the equivalent norms of step 3.1.
The shear is defined on the box $Q_{r}$ and the identities in step 2.1 use only the local graph representation; no global parametrisation of $\partial Ω$ is asserted.

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Boundary Schauder estimate for the Dirichlet problem

Statement

Assume the Axiom of Choice and Countable Choice. Let $n \geq 2$ , $0 < α < 1$ , let $Ω$ be a bounded $C^{2, α}$ domain, and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic on $\overset{ˉ}{Ω}$ with constants $λ, Λ$ , $[A]_{0, α; Ω} \leq K$ , and $b, c \in C^{0, α} (\overset{ˉ}{Ω})$ with $∥ b ∥_{C^{0, α}} + ∥ c ∥_{C^{0, α}} \leq M$ . Let $u \in C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ , $g \in C^{2, α} (\overset{ˉ}{Ω})$ and $f \in C^{0, α} (\overset{ˉ}{Ω})$ satisfy $L u = f$ in $Ω$ and $u = g$ on $\partial Ω$ . Then $u \in C^{2, α} (\overset{ˉ}{Ω})$ and $∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C (∥ u ∥_{C^{0} (Ω)} + ∥ f ∥_{C^{0, α} (\overset{ˉ}{Ω})} + ∥ g ∥_{C^{2, α} (\overset{ˉ}{Ω})}),$ where $C = C (n, α, λ, Λ, K, M, Ω)$ depends on $Ω$ only through a finite $C^{2, α}$ chart atlas and its radii. The coefficient assumptions include the H"older norms needed for the lower-order products, and no solvability is asserted: the estimate is a regularity statement for the given classical solution.

Facts & Assumptions

Given: the Axiom of Choice and ${A C}_{ω}$ , $n \geq 2$ , $0 < α < 1$ , the bounded $C^{2, α}$ domain $Ω$ , the operator $L$ with the stated bounds, and $u \in C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ , $g \in C^{2, α} (\overset{ˉ}{Ω})$ , $f \in C^{0, α} (\overset{ˉ}{Ω})$ with $L u = f$ in $Ω$ and $u = g$ on $\partial Ω$ .

[A1]

The Axiom of Choice is assumed for the quoted boundary regularity and estimate inputs; Countable Choice is inherited by the measure and local estimate interfaces. The chart cover itself involves only finitely many selections. (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The classes $C^{2, α} (\overset{ˉ}{Ω})$ and $C^{0, α} (\overset{ˉ}{Ω})$ are the boundary-extension classes of Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains: an element of $C_{b}^{2, α} (Ω)$ lies in $C^{2, α} (\overset{ˉ}{Ω})$ when its derivatives through order $2$ extend continuously, and then $∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})} = \sum_{j = 0}^{2} \sup_{Ω} \max_{∣ β ∣ = j} ∣ D^{β} u ∣ + [D^{2} u]_{0, α; Ω}$ . On a ball $B_{R} (x_{0})$ the scaled quantity is $∥ w ∥_{2, α; B_{R} (x_{0})}^{*} = \sum_{j = 0}^{2} R^{j} \max_{∣ β ∣ = j} \sup_{B_{R}} ∣ D^{β} w ∣ + R^{2 + α} \max_{∣ β ∣ = 2} [D^{β} w]_{0, α; B_{R}}$ and $∥ h ∥_{0, α; B_{R}}^{*} = \sup_{B_{R}} ∣ h ∣ + R^{α} [h]_{0, α; B_{R}}$ , with the analogous formulas on a half-box; the scaled quantities dominate each of their defining terms. (Local Hölder and scaled C-two-alpha norms on balls)

[F2]

Interior Schauder estimate (Interior Schauder estimate for uniformly elliptic equations): let $B_{R} (x_{0}) \subseteq R^{n}$ , let $L$ be uniformly elliptic on $B_{R} (x_{0})$ with constants $λ, Λ$ , $[A]_{0, α; B_{R} (x_{0})} \leq K$ and $∥ b ∥_{C^{0, α}} + ∥ c ∥_{C^{0, α}} \leq M$ there. If $w \in C^{2} (B_{R} (x_{0})) \cap L^{\infty} (B_{R} (x_{0}))$ satisfies $L w = F$ pointwise with $F \in C^{0, α} (B_{R} (x_{0}))$ , then $w \in C^{2, α} (B_{R / 2} (x_{0}))$ and $∥ w ∥_{2, α; B_{R / 2} (x_{0})}^{*} \leq C (∥ w ∥_{\infty; B_{R} (x_{0})} + R^{2} ∥ F ∥_{\infty; B_{R} (x_{0})} + R^{2 + α} [F]_{0, α; B_{R} (x_{0})})$ with $C$ depending only on $n, α, λ, Λ$ and the dimensionless coefficient bounds $R^{α} K + R ∥ b ∥_{\infty} + R^{1 + α} [b]_{0, α} + R^{2} ∥ c ∥_{\infty} + R^{2 + α} [c]_{0, α}$ .

[F3]

Boundary flattening ( $C^{2, α}$ boundary flattening preserves the nondivergence structure, ellipticity and Hölder norms): for every boundary point there are $r > 0$ and a shear $Ψ$ with $Ψ (Q_{r}^{+}) = U \cap Ω$ and $Ψ (Q_{r}^{'} \times {0}) = U \cap \partial Ω$ , where $Q_{r} = Q_{r}^{'} \times (- r, r)$ and $Q_{r}^{+} = Q_{r}^{'} \times (0, r)$ ; by shrinking inside a larger graph chart, take $Ψ$ defined on a neighborhood of $\overline{Q_{r}}$ and $Ψ (\overline{Q_{r}^{+}}) \subset \overset{ˉ}{Ω}$ . Composition with $Ψ$ gives equivalent $C^{2, α}$ norms on the closures of $U \cap Ω$ and $Q_{r}^{+}$ with constants depending only on the chart, and if $L$ has $C^{0, α}$ coefficients on $U$ then its pullback $\tilde{L} = {\tilde{a}}^{i j} \partial_{i} \partial_{j} + {\tilde{b}}^{i} \partial_{i} + \tilde{c}$ under $v = u \circ Ψ$ has $C^{0, α}$ coefficients with ellipticity constants $\tilde{λ} = λ ∥ D Ψ ∥_{\infty}^{- 2}$ , $\tilde{Λ} = Λ ∥ D Ψ^{- 1} ∥_{\infty}^{2}$ and coefficient bounds controlled by the chart and by $K, M$ . For coefficients given only on $\overset{ˉ}{Ω}$ , extend each field to the chart image by $q (Ψ (y^{'}, y_{n})) : = q (Ψ (y^{'}, ∣ y_{n} ∣))$ . Reflection and the bi-Lipschitz shear preserve its H"older bounds, and evaluating the same original matrix preserves ellipticity. Thus the lemma applies on the ambient patch. The shear is available at each boundary point of a bounded $C^{2, α}$ domain because such a domain carries graph charts of class $C^{2, α}$ over which the shear of the lemma straightens the boundary. (Bounded C^k domains and boundary charts)

[F4]

Quoted boundary inputs with their distinct hypotheses. Gilbarg–Trudinger, Elliptic Partial Differential Equations of Second Order (2001), Lemma 6.18 and Theorem 6.19, printed p.111, give local $C^{2, α}$ regularity up to a $C^{2, α}$ boundary portion for $u \in C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ , with Hölder coefficients and forcing and $C^{2, α}$ boundary values; no sign condition on $c$ is imposed. Apply this first at the flat face. Then Simon, Lecture 12 Theorem 2', printed pp.134–135, gives the a priori estimate on a smaller half-patch. Covering the smaller half-box by such patches and interior balls gives $∥ v ∥_{2, α; Q_{r / 2}^{+}}^{*} \leq C (∥ v ∥_{\infty; Q_{r}^{+}} + r^{2} ∥ h ∥_{0, α; Q_{r}^{+}}^{*})$ for zero flat-face data. Rescaling includes $r^{α} \tilde{K}, r ∥ \tilde{b} ∥_{\infty}, r^{1 + α} [\tilde{b}]_{α}, r^{2} ∥ \tilde{c} ∥_{\infty}, r^{2 + α} [\tilde{c}]_{α}$ in $C$ . These are explicit literature inputs; Wang Theorem 1' alone assumes $C^{2}$ up to the flat face and does not supply the upgrade.

Proof

technique · direct

1.1givenF1algebra

Reduction to zero boundary values. Put $w : = u - g$ on $Ω$ . Since $g \in C^{2, α} (\overset{ˉ}{Ω}) \subseteq C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ , the function $w$ lies in $C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ , vanishes on $\partial Ω$ , and satisfies $L w = F$ pointwise in $Ω$ with $F : = f - L g$ . The product inequality $[a b]_{α} \leq ∥ a ∥_{\infty} [b]_{α} + [a]_{α} ∥ b ∥_{\infty}$ gives $[L g]_{0, α; Ω} \leq C_{n} (Λ [D^{2} g]_{0, α; Ω} + K ∥ D^{2} g ∥_{\infty; Ω} + ∥ b ∥_{\infty} [D g]_{0, α; Ω} + [b]_{0, α; Ω} ∥ D g ∥_{\infty; Ω} + ∥ c ∥_{\infty} [g]_{0, α; Ω} + [c]_{0, α; Ω} ∥ g ∥_{\infty; Ω})$ and $\sup_{Ω} ∣ L g ∣ \leq C_{n} (Λ + M) ∥ g ∥_{C^{2, α} (Ω)}$ ; in each flattened half-box the lower-field seminorms are bounded by the derivative suprema using integration along segments, and a finite cover plus the separation bound controls pairs in different charts; hence $F \in C^{0, α} (\overset{ˉ}{Ω})$ with $∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})} \leq ∥ f ∥_{C^{0, α} (\overset{ˉ}{Ω})} + C (n, α, Λ, K, M, Ω) ∥ g ∥_{C^{2, α} (\overset{ˉ}{Ω})}$ , and $\sup_{Ω} ∣ w ∣ \leq ∥ u ∥_{C^{0} (Ω)} + ∥ g ∥_{C^{2, α} (\overset{ˉ}{Ω})}$ . It thus suffices to show that every such zero-boundary $w$ with $L w = F \in C^{0, α} (\overset{ˉ}{Ω})$ satisfies $w \in C^{2, α} (\overset{ˉ}{Ω})$ and $∥ w ∥_{C^{2, α} (Ω)} \leq C (∥ w ∥_{\infty; Ω} + ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})})$ with $C = C (n, α, λ, Λ, K, M, Ω)$ ; adding $g$ back gives the statement.

1.2F3givenconstructA1

A finite relative cover with a Lebesgue number. By [F3], for every $x \in \partial Ω$ there is a flattened chart $Ψ_{x} : Q_{r_{x}} \to U_{x}$ with $Ψ_{x} (Q_{r_{x}}^{+}) = U_{x} \cap Ω$ and $Ψ_{x} (Q_{r_{x}}^{'} \times {0}) = U_{x} \cap \partial Ω$ . Put ${\hat{V}}_{x} : = Ψ_{x} (Q_{r_{x} / 2})$ , an ambient open neighborhood of $x$ that includes points on both sides of the flattened boundary. These sets cover $\partial Ω$ ; compactness gives finitely many, indexed by $ℓ = 1, \dots, N$ , and their union $V$ is an open neighborhood of $\partial Ω$ in $R^{n}$ . Thus some $δ > 0$ satisfies ${y \in Ω : dist (y, \partial Ω) < δ} \subseteq V$ . The set $Ω_{δ} : = {y \in Ω : dist (y, \partial Ω) \geq δ}$ is compact in $Ω$ . With $R : = δ / 4$ , its balls $B_{R} (z)$ have doubled balls inside $Ω$ , and compactness yields finitely many centres $z_{1}, \dots, z_{P}$ whose inner balls cover $Ω_{δ}$ . The finite family consisting of the ambient open sets ${\hat{V}}_{ℓ}$ and the balls $B_{R} (z_{j})$ covers $\overset{ˉ}{Ω}$ relative to $\overset{ˉ}{Ω}$ . Let $δ_{0} > 0$ be a Lebesgue number for this relative open cover, so every pair $x, y \in \overset{ˉ}{Ω}$ with $∣ x - y ∣ < δ_{0}$ lies in one common member.

1.3F3F4F1givenalgebra

Estimates on the boundary members. Fix $ℓ \in {1, \dots, N}$ , write $Ψ = Ψ_{ℓ}$ , $r = r_{ℓ}$ and let $\tilde{w} : = w \circ Ψ$ on $Q_{r}^{+}$ , with pullback operator $\tilde{L}$ as in [F3]. Then $\tilde{w} \in C^{2} (Q_{r}^{+}) \cap C^{0} (\overline{Q_{r}^{+}})$ because $Ψ$ is a $C^{2, α}$ diffeomorphism of a neighbourhood of $\overline{Q_{r}^{+}}$ onto a neighbourhood of $\overline{U_{ℓ} \cap Ω}$ and $w \in C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ ; moreover $\tilde{w} = 0$ on the flat face, since that face maps onto $U_{ℓ} \cap \partial Ω$ where $w = 0$ . The pullback satisfies $\tilde{L} \tilde{w} = \tilde{h}$ pointwise in $Q_{r}^{+}$ with $\tilde{h} : = F \circ Ψ$ , which lies in $C^{0, α} (Q_{r}^{+}) \cap C^{0} (\overline{Q_{r}^{+}})$ by composition, and $∥ \tilde{h} ∥_{0, α; Q_{r}^{+}}^{*} \leq C_{ℓ} ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})}$ with $C_{ℓ}$ depending on the chart. Applying the quoted boundary estimate [F4] to $\tilde{w}$ gives $∥ \tilde{w} ∥_{2, α; Q_{r / 2}^{+}}^{*} \leq C_{ℓ}^{'} (∥ w ∥_{\infty; Ω} + ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})}),$ with $C_{ℓ}^{'}$ depending on $n, α$ , the ellipticity constants and coefficient bounds of $\tilde{L}$ (hence on the chart and on $λ, Λ, K, M$ ). By the norm equivalence of [F3], the last display controls $\sup ∣ D^{β} w ∣$ for $∣ β ∣ \leq 2$ and $[D^{2} w]_{0, α}$ over $Ψ (Q_{r / 2}^{+}) = U_{ℓ} \cap V_{ℓ}$ with $V_{ℓ} : = Ψ_{ℓ} (Q_{r_{ℓ} / 2}^{+})$ ; finitely many charts give one constant $C_{2}$ .

2.1F2F1step 1.2algebra

Estimates on the interior members. Fix $j \in {1, \dots, P}$ and let $B_{2 R} (z_{j}) \subseteq Ω$ . On $B_{2 R} (z_{j})$ the function $w$ is of class $C^{2}$ and bounded, and $L w = F$ pointwise with $F \in C^{0, α} (B_{2 R} (z_{j}))$ ; the coefficient bounds $[A]_{0, α} \leq K$ and $∥ b ∥_{C^{0, α}} + ∥ c ∥_{C^{0, α}} \leq M$ hold there. By [F2] with radius $2 R$ , $∥ w ∥_{2, α; B_{R} (z_{j})}^{*} \leq C_{j} (∥ w ∥_{\infty; Ω} + ∥ F ∥_{\infty; Ω} + [F]_{0, α; Ω}),$ where $C_{j}$ depends on $n, α, λ, Λ$ and the dimensionless bounds formed with the radius $2 R$ ; as there are finitely many $j$ and all radii are comparable to $δ$ , the numbers $C_{j}$ are bounded by a constant $C_{1}$ depending on $n, α, λ, Λ, K, M$ and on the cover. In particular $\sup_{B_{R} (z_{j})} ∣ D^{β} w ∣ \leq R^{- ∣ β ∣} ∥ w ∥_{2, α; B_{R} (z_{j})}^{*}$ for $∣ β ∣ \leq 2$ and $[D^{2} w]_{0, α; B_{R} (z_{j})} \leq R^{- 2 - α} ∥ w ∥_{2, α; B_{R} (z_{j})}^{*}$ .

3.1step 1.2step 2.1step 1.3F1algebra

Summation and conclusion. The interior balls $B_{R} (z_{j})$ together with the boundary-chart sets ${\hat{V}}_{ℓ} \cap \overset{ˉ}{Ω}$ cover $\overset{ˉ}{Ω}$ ; on each boundary-chart set, its portion in $\overset{ˉ}{Ω}$ lies in $Ψ_{ℓ} (\overline{Q_{r_{ℓ} / 2}^{+}})$ , where step 1.3 supplies the estimate. Thus for every $∣ β ∣ \leq 2$ , $\sup_{Ω} ∣ D^{β} w ∣ \leq \sum_{j} \sup_{B_{R} (z_{j})} ∣ D^{β} w ∣ + \sum_{ℓ} \sup_{{\hat{V}}_{ℓ} \cap Ω} ∣ D^{β} w ∣ \leq C_{3} (∥ w ∥_{\infty; Ω} + ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})})$ by steps 2.1 and 1.3. For the H"older seminorm let $x, y \in Ω$ , $x \neq y$ . If $∣ x - y ∣ < δ_{0}$ , step 1.2 gives a member of the relative cover containing both points, and steps 2.1 or 1.3 bound the corresponding difference quotient; if $∣ x - y ∣ \geq δ_{0}$ , then $∣ D^{2} w (x) - D^{2} w (y) ∣ / ∣ x - y ∣^{α} \leq 2 δ_{0}^{- α} \sup_{Ω} ∣ D^{2} w ∣$ , already controlled. Therefore $∥ w ∥_{C^{2, α} (Ω)} = \sum_{j = 0}^{2} \sup_{Ω} \max_{∣ β ∣ = j} ∣ D^{β} w ∣ + [D^{2} w]_{0, α; Ω} \leq C (∥ w ∥_{\infty; Ω} + ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})})$ with $C = C (n, α, λ, Λ, K, M, Ω)$ . Since the boundary-chart sets are ambient neighborhoods of each boundary point and the controlled coordinate functions are $C^{2, α}$ up to the flat face, the derivatives $D^{β} w$ for $∣ β ∣ \leq 2$ extend continuously to $\overset{ˉ}{Ω}$ . Hence $w \in C^{2, α} (\overset{ˉ}{Ω})$ with the same norm over $Ω$ , as required by [F1].

4.1step 1.1step 3.1F2F4given∎

The estimate for $u$ . By step 1.1 and step 3.1, $∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq ∥ w ∥_{C^{2, α} (Ω)} + ∥ g ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C (∥ u ∥_{C^{0} (Ω)} + ∥ f ∥_{C^{0, α} (\overset{ˉ}{Ω})} + ∥ g ∥_{C^{2, α} (\overset{ˉ}{Ω})})$ after enlarging $C$ to absorb the constants of step 1.1; the constant depends on $n, α, λ, Λ, K, M$ and on the finite cover, i.e. on $Ω$ only through its $C^{2, α}$ charts and radii. The strict range $0 < α < 1$ enters through the quoted boundary estimate [F4] and the interior estimate [F2]; the Dirichlet condition enters through $w = 0$ on the boundary, which is exactly the zero-data hypothesis of [F4] on each flat face. No solvability, compactness of the operator or boundary regularity beyond the $C^{2, α}$ chart hypothesis is used, and it is the Dirichlet condition and the $C^{2, α}$ boundary that make the estimate possible at every boundary point.

Remarks

The regularity upgrade and the subsequent a priori estimate have different hypotheses, explicitly separated in [F4].
The cover argument is the same one used for the interior estimate, run on the compact closure; the Lebesgue number replaces the partition of unity, which is why the covering selections are finite; the quoted analytic inputs are used under [A1].
The estimate is stated with $∥ u ∥_{C^{0} (Ω)}$ rather than $∥ u ∥_{C^{2, α}}$ on the right, so it is a genuine a priori bound. The interpolation lemma of the interior argument is not needed in this form of the proof, because the quoted local a priori estimate already handles the lower-order terms; the intermediate-derivative terms are not produced by cutoffs here.

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The method of continuity for a uniformly estimated affine family of bounded operators

Statement

Assume Countable Choice. Let $X, Y$ be Banach spaces over the same field, let $L_{0}, L_{1} \in B (X, Y)$ and put $L_{t} : = (1 - t) L_{0} + t L_{1}$ for $t \in [0, 1]$ . Assume (i) $L_{0}$ is bijective; (ii) there is $0 \leq C < \infty$ with the uniform a priori estimate $∥ x ∥_{X} \leq C ∥ L_{t} x ∥_{Y}$ for every $t \in [0, 1]$ and every $x \in X$ . Then $L_{t}$ is bijective for every $t \in [0, 1]$ , and $∥ L_{t}^{- 1} ∥_{Y \to X} \leq C$ for every $t$ . No compactness or reflexivity hypothesis is used: the uniform estimate alone makes the bijectivity set closed, and the Neumann series makes it open.

Facts & Assumptions

Given: ${A C}_{ω}$ , Banach spaces $X, Y$ over the same field, operators $L_{0}, L_{1} \in B (X, Y)$ , the affine family $L_{t} = (1 - t) L_{0} + t L_{1}$ , and the hypotheses (i) $L_{0}$ bijective, (ii) $0 \leq C < \infty$ and $∥ x ∥ \leq C ∥ L_{t} x ∥$ for all $t \in [0, 1]$ , all $x \in X$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ , used through the sequential completeness conventions of the Banach spaces. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

A Banach space is a normed space whose norm metric is complete, so every Cauchy sequence converges; limits in a metric space are unique. (Banach space, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ )

[F2]

$B (X, Y)$ consists of the bounded linear maps $X \to Y$ , with pointwise operations, and $∥ T x ∥ \leq ∥ T ∥ ∥ x ∥$ ; the operator norm is subadditive and homogeneous, so $∥ (1 - t) L_{0} + t L_{1} ∥ \leq (1 - t) ∥ L_{0} ∥ + t ∥ L_{1} ∥$ for $t \in [0, 1]$ , and for $T \in B (X, Y)$ , $S \in B (Y, Z)$ one has $∥ S T ∥ \leq ∥ S ∥ ∥ T ∥$ . (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces, Composition satisfies |ST|\le|S|,|T|)

[F3]

If $X$ is a Banach space and $R \in B (X)$ with $∥ R ∥ < 1$ , then $I + R$ is invertible with inverse $\sum_{n \geq 0} (- R)^{n}$ and $∥ (I + R)^{- 1} ∥ \leq (1 - ∥ R ∥)^{- 1}$ ; if $A \in B (X, Y)$ is invertible and $E \in B (X, Y)$ satisfies $∥ A^{- 1} E ∥ < 1$ , then $A + E$ is invertible. (Neumann series and small perturbations of bounded inverses)

Proof

technique · direct

1.1givenF2algebra

Injectivity and uniform lower bound. Fix $t \in [0, 1]$ . If $L_{t} x = 0$ then [ii] gives $∥ x ∥ \leq C ∥ L_{t} x ∥ = 0$ , so $x = 0$ : every $L_{t}$ is injective. Moreover [ii] says exactly that $∥ L_{t}^{- 1} y ∥ \leq C ∥ y ∥$ for every $y$ in the range $L_{t} (X)$ , so whenever $L_{t}$ is surjective its inverse is bounded with norm at most $C$ .

2.1step 1.1F1F2algebra

The bijectivity set is closed in $[0, 1]$ . Let $t_{j} \to t$ in $[0, 1]$ with every $L_{t_{j}}$ bijective, let $f \in Y$ and put $u_{j} : = L_{t_{j}}^{- 1} f$ . Then $∥ u_{j} ∥ \leq C ∥ f ∥$ by step 1.1, and for all $j, k$ the identity $L_{t_{k}} (u_{j} - u_{k}) = L_{t_{k}} u_{j} - f = (L_{t_{k}} - L_{t_{j}}) u_{j} = (t_{k} - t_{j}) (L_{1} - L_{0}) u_{j}$ together with [ii] and [F2] gives $∥ u_{j} - u_{k} ∥ \leq C ∥ L_{t_{k}} (u_{j} - u_{k}) ∥ \leq C^{2} ∣ t_{k} - t_{j} ∣ ∥ L_{1} - L_{0} ∥ ∥ f ∥$ , so $(u_{j})$ is Cauchy in $X$ ; by [F1] it converges to some $u \in X$ . Since $∥ L_{t} - L_{t_{j}} ∥ \leq ∣ t - t_{j} ∣ ∥ L_{1} - L_{0} ∥$ by [F2] and the sequence $(∥ u_{j} ∥)$ is bounded by $C ∥ f ∥$ , $∥ L_{t} u - f ∥ \leq ∥ (L_{t} - L_{t_{j}}) u ∥ + ∥ L_{t_{j}} (u - u_{j}) ∥ + ∥ L_{t_{j}} u_{j} - f ∥ \leq ∣ t - t_{j} ∣ ∥ L_{1} - L_{0} ∥ ∥ u ∥ + ((1 - t_{j}) ∥ L_{0} ∥ + t_{j} ∥ L_{1} ∥) ∥ u - u_{j} ∥$ , and both terms tend to $0$ ; hence $L_{t} u = f$ . So $L_{t}$ is surjective, injective by step 1.1, and therefore bijective with $∥ L_{t}^{- 1} ∥ \leq C$ by step 1.1. This shows that a limit of bijective parameters is bijective, that is, the bijectivity set $I : = {t \in [0, 1] : L_{t} bijective}$ is closed in $[0, 1]$ .

2.2step 1.1F2F3casesalgebra

The bijectivity set is open in $[0, 1]$ . Let $t \in I$ . If $C = 0$ , the estimate implies $X = {0}$ , and bijectivity of $L_{0}$ implies $Y = {0}$ , so every $L_{s}$ is the unique bijection and $I = [0, 1]$ . Assume $C > 0$ . If $L_{1} = L_{0}$ then $L_{s} = L_{t}$ for every $s$ and the claim is trivial, so assume $∥ L_{1} - L_{0} ∥ > 0$ and let $s \in [0, 1]$ satisfy $∣ s - t ∣ < 1 / (C ∥ L_{1} - L_{0} ∥)$ . Write $L_{s} = L_{t} + (s - t) (L_{1} - L_{0}) = L_{t} (I + L_{t}^{- 1} (s - t) (L_{1} - L_{0}))$ , where $L_{t}^{- 1} \in B (Y, X)$ has norm at most $C$ by step 1.1. Since $∥ L_{t}^{- 1} (s - t) (L_{1} - L_{0}) ∥ \leq C ∣ s - t ∣ ∥ L_{1} - L_{0} ∥ < 1$ , the Neumann series [F3] makes $I + L_{t}^{- 1} (s - t) (L_{1} - L_{0})$ invertible on $X$ with inverse in $B (X)$ ; composing with the bijection $L_{t}$ shows that $L_{s}$ is bijective, with inverse $(I + L_{t}^{- 1} (s - t) (L_{1} - L_{0}))^{- 1} L_{t}^{- 1} \in B (Y, X)$ and norm at most $C (1 - C ∣ s - t ∣ ∥ L_{1} - L_{0} ∥)^{- 1}$ . Hence $I$ is open in $[0, 1]$ .

3.1step 1.1step 2.1step 2.2A1given∎

Conclusion. $I$ is nonempty because $0 \in I$ by (i), and it is open and closed in $[0, 1]$ by steps 2.1 and 2.2. Suppose $I \neq [0, 1]$ , and let $t : = \sup {y \in [0, 1] : [0, y] \subseteq I}$ , a set that contains $0$ and is nonempty. For $y < t$ one has $[0, y] \subseteq I$ , and closedness of $I$ gives $t \in I$ (if $t = 0$ , use $0 \in I$ ). If $t = 1$ this already gives $I = [0, 1]$ , a contradiction; so $t < 1$ ; openness of $I$ then gives $0 < δ \leq 1 - t$ with $(t - δ, t + δ) \cap [0, 1] \subseteq I$ , so $[0, t + δ / 2] \subseteq I$ , contradicting the definition of $t$ . Hence $I = [0, 1]$ : every $L_{t}$ is bijective, and $∥ L_{t}^{- 1} ∥ \leq C$ for every $t$ by step 1.1. No compactness, reflexivity or separability of $X$ or $Y$ was used anywhere; the only completeness used is that of $X$ in step 2.1 and the only choice principle is the sequential convention of [A1].

Remarks

If the uniform estimate [ii] holds only for $t$ in a subset $A \subseteq [0, 1]$ , the argument shows that the bijectivity set is relatively open and relatively closed in $A$ ; the interval $[0, 1]$ is used only to run the endpoint propagation in step 3.1.
The uniform lower bound controls the inverses and the Cauchy sequence in the closedness proof; both openness and closedness also use that $t \mapsto L_{t}$ is affine and hence Lipschitz with constant $∥ L_{1} - L_{0} ∥$ .

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Global $W^{2, p}$ estimate for the Laplacian on Euclidean space

Statement

Assume Countable Choice. Let $n \geq 2$ and $1 < p < \infty$ . Then there is $C = C (n, p) < \infty$ such that every $u \in C_{c}^{\infty} (R^{n})$ satisfies $∥ D^{2} u ∥_{L^{p} (R^{n})} \leq C ∥ Δ u ∥_{L^{p} (R^{n})}, ∥ \partial_{i} \partial_{j} u ∥_{L^{p} (R^{n})} \leq C ∥ Δ u ∥_{L^{p} (R^{n})},$ for all $i, j$ , where $∥ D^{2} u ∥_{L^{p}} : = \max_{∣ β ∣ = 2} ∥ D^{β} u ∥_{L^{p}}$ ; by density the bound extends to every $u \in W^{2, p} (R^{n})$ (for which $Δ u \in L^{p}$ holds automatically). The constant may be taken as the square of the Riesz-transform bound of The Riesz transforms are bounded on Lp, hence is finite throughout $1 < p < \infty$ , including $p = 2$ . No sharp endpoint growth rate is claimed.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 2$ , $1 < p < \infty$ , and a function $u$ that is either in $C_{c}^{\infty} (R^{n})$ or, in the density step, in $W^{2, p} (R^{n})$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the choice-qualified Riesz-transform, Fourier and Sobolev interfaces below. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The Riesz transforms are the $L^{2}$ operators $R_{j} = F_{2}^{- 1} M_{m_{j}} F_{2}$ with $m_{j} (ξ) = - i ξ_{j} / ∣ ξ ∣$ for $ξ \neq 0$ and $m_{j} (0) = 0$ , and for every $1 < p < \infty$ each $R_{j}$ extends uniquely to a bounded operator on $L^{p} (R^{n}; C)$ with norm at most $C_{n, p}$ ; the bound of The Riesz transforms are bounded on Lp is $\leq C_{n, p} (1 + ∣ S^{n - 1} ∣ 2^{- 1} C_{n}) \max (p, (p - 1)^{- 1})$ ; this is an upper bound, not a lower bound on the operator norm. (Riesz transforms on Euclidean space, Exact L2 Fourier multiplier norm)

[F2]

The unitary Plancherel transform $F_{2}$ is complex-linear, isometric and injective on $L^{2} (R^{n}; C)$ ; the negative-sign $2 π$ -normalized distributional transform equals $F_{2}$ on $L^{2}$ classes in the sense that $F u_{f} = u_{F_{2} f}$ , and it satisfies $F (\partial^{α} u) = (2 π i ξ)^{α} F u$ for tempered distributions. (Plancherel theorem, Fourier transform agrees with l one and plancherel transforms, Fourier differentiation and multiplication identities on tempered distributions)

[F3]

Compactly supported smooth functions are dense in $W^{2, p} (R^{n})$ for finite $p$ , and the Sobolev norm is the $p$ -sum of the $L^{p}$ norms of the weak derivatives; for $u \in W^{2, p}$ all weak second derivatives and hence $Δ u = \sum_{i} \partial_{i}^{2} u$ lie in $L^{p}$ . (Compactly supported smooth functions are dense in W^{k,p}(R^n), Integer-order Sobolev spaces and their norms)

Proof

technique · direct

1.1F1F2givenalgebra

Fourier identification. Let $u \in C_{c}^{\infty} (R^{n})$ and put $f : = - Δ u \in C_{c}^{\infty} (R^{n})$ . Since $u$ is smooth, the classical identity $\hat{\partial_{i} \partial_{j} u} = (2 π i ξ_{i}) (2 π i ξ_{j}) \hat{u}$ and the distributional Fourier calculus of [F2] give, as tempered distributions, $F (\partial_{i} \partial_{j} u) = (2 π i ξ_{i}) (2 π i ξ_{j}) F u = - 4 π^{2} ξ_{i} ξ_{j} F u$ and $F (- Δ u) = 4 π^{2} ∣ ξ ∣^{2} F u$ ; on $L^{2}$ classes these equal $F_{2} (\partial_{i} \partial_{j} u)$ and $F_{2} f$ respectively by the agreement statement of [F2]. Since $f \in L^{2}$ and $R_{j}$ is the $L^{2}$ multiplier by $m_{j}$ , the composition satisfies $R_{i} R_{j} f = F_{2}^{- 1} (m_{i} m_{j} F_{2} f)$ and, on ${ξ \neq 0}$ , $m_{i} m_{j} \cdot 4 π^{2} ∣ ξ ∣^{2} = (- i ξ_{i} / ∣ ξ ∣) (- i ξ_{j} / ∣ ξ ∣) 4 π^{2} ∣ ξ ∣^{2} = - 4 π^{2} ξ_{i} ξ_{j}$ . Plancherel injectivity [F2] therefore gives the $L^{2}$ identity $\partial_{i} \partial_{j} u = R_{i} R_{j} f = R_{i} R_{j} (- Δ u)$ .

2.1step 1.1F1algebra

$L^{p}$ bound for smooth compactly supported data. For $u \in C_{c}^{\infty}$ the function $f = - Δ u$ lies in $C_{c}^{\infty} \subset L^{p} \cap L^{2}$ , so both operators in step 1.1 are defined on $L^{p}$ and the identity holds a.e.; using twice the $L^{p}$ bound of [F1], $∥ \partial_{i} \partial_{j} u ∥_{L^{p}} = ∥ R_{i} R_{j} f ∥_{L^{p}} \leq C_{n, p}^{2} ∥ f ∥_{L^{p}} = C_{n, p}^{2} ∥ Δ u ∥_{L^{p}}$ . Taking the maximum over $i, j$ gives $∥ D^{2} u ∥_{L^{p}} \leq C_{n, p}^{2} ∥ Δ u ∥_{L^{p}}$ for every $u \in C_{c}^{\infty} (R^{n})$ .

3.1step 2.1F3algebra

Density. Let $u \in W^{2, p} (R^{n})$ and let $u_{k} \in C_{c}^{\infty} (R^{n})$ satisfy $u_{k} \to u$ in $W^{2, p} (R^{n})$ , which exists by [F3]. Then $Δ u_{k} \to Δ u$ and $\partial_{i} \partial_{j} u_{k} \to \partial_{i} \partial_{j} u$ in $L^{p}$ by the definition of the Sobolev norm [F3], and applying step 2.1 to $u_{k}$ and passing to the limit gives $∥ D^{2} u ∥_{L^{p}} \leq C_{n, p}^{2} ∥ Δ u ∥_{L^{p} (R^{n})}$ and the same bound for each $\partial_{i} \partial_{j} u$ . Since $Δ u \in L^{p}$ holds automatically for $W^{2, p}$ classes by [F3], the inequality applies to every such class.

4.1step 2.1step 3.1F1A1given∎

Conclusion and constants. The two displayed inequalities hold with $C = C_{n, p}^{2}$ , which is finite for every $1 < p < \infty$ by [F1]. Squaring an upper bound supplies an upper bound only; no sharp growth rate or endpoint estimate is inferred. The proof uses the Riesz-transform $L^{p}$ theory, whose choice assumption is the Countable Choice of [A1] together with those of the Fourier interfaces; no compactness, no extension operator and no maximal-function argument is used.

Remarks

The identity $\partial_{i} \partial_{j} u = R_{i} R_{j} (- Δ u)$ is the multiplier form of the classical relation $ξ_{i} ξ_{j} = (ξ_{i} ξ_{j} / ∣ ξ ∣^{2}) ∣ ξ ∣^{2}$ ; the cancellation at $ξ = 0$ is immaterial because single points are Lebesgue null.
The estimate is the $L^{p}$ counterpart of the Schauder estimate of this page: both control second derivatives by the Laplacian/operator, but the $L^{p}$ scale accepts merely $L^{p}$ data and its constant degenerates at $p = 1$ and $p = \infty$ .

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$L^{p}$ interpolation absorption of first derivatives by second derivatives

Statement

Assume Countable Choice. Let $n \geq 1$ and $1 < p < \infty$ , and for a function with the relevant weak derivatives write $∥ D u ∥_{L^{p}} : = \max_{∣ β ∣ = 1} ∥ D^{β} u ∥_{L^{p}}$ and $∥ D^{2} u ∥_{L^{p}} : = \max_{∣ β ∣ = 2} ∥ D^{β} u ∥_{L^{p}}$ , equivalent to the sum-form Sobolev norms of Integer-order Sobolev spaces and their norms up to constants depending on $n$ . For every $ε > 0$ there is $C = C (n, p, ε) < \infty$ such that every $u \in W^{2, p} (R^{n})$ satisfies $∥ D u ∥_{L^{p} (R^{n})} \leq ε ∥ D^{2} u ∥_{L^{p} (R^{n})} + C ∥ u ∥_{L^{p} (R^{n})} .$ The scaled form on balls, with the norm over the doubled ball on the right, is $∥ D u ∥_{L^{p} (B_{R} (x_{0}))} \leq ε R ∥ D^{2} u ∥_{L^{p} (B_{2 R} (x_{0}))} + C R^{- 1} ∥ u ∥_{L^{p} (B_{2 R} (x_{0}))} (u \in W^{2, p} (B_{2 R} (x_{0}))),$ with $C$ independent of $R$ and $x_{0}$ . This is the absorption inequality used in the frozen-coefficient $W^{2, p}$ estimates. The result is asserted for the strict range $1 < p < \infty$ only; the form with the same ball on both sides is not claimed here.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , $1 0$ , the Euclidean ball $B_{R} (x_{0}) = {x : ∣ x - x_{0} ∣ < R}$ , and the multiplier and Sobolev conventions below.

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the choice-qualified Fourier, multiplier, Sobolev and measure interfaces cited below. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

A measurable $m$ is a Mihlin symbol when $m = m_{0}$ a.e. for some $m_{0} \in C^{q} (R^{n} ∖ {0})$ , $q = ⌊ n / 2 ⌋ + 1$ , with $∣ \partial^{α} m_{0} (ξ) ∣ \leq C_{α} ∣ ξ ∣^{- ∣ α ∣}$ for $∣ α ∣ \leq q$ and $ξ \neq 0$ . Every Mihlin symbol is an $L^{p}$ multiplier for $1 < p < \infty$ with $∥ m ∥_{M_{p}} \leq C_{n} \max (p, (p - 1)^{- 1}) (A + ∥ m ∥_{\infty})$ , $A = \max_{∣ α ∣ \leq q} C_{α}$ , in the multiplier conventions of the cited items. (Mihlin smoothness convention above half the dimension, The Mihlin–Hörmander Fourier multiplier theorem, Lp Fourier multiplier and its norm, Translation-invariant Fourier multiplier on the Schwartz core)

[F2]

The Fourier transform is the negative-sign $2 π$ -normalized transform, an automorphism of $S^{'} (R^{n})$ that is injective on tempered distributions, with $F (\partial^{α} u) = (2 π i ξ)^{α} F u$ for every multi-index $α$ . (Fourier differentiation and multiplication identities on tempered distributions, Fourier transform is a topological automorphism of tempered distributions)

[F3]

For finite $p$ the Sobolev norm of Integer-order Sobolev spaces and their norms is the $p$ -sum of the $L^{p}$ norms of the weak derivatives, and $L^{p} (R^{n})$ norms obey the triangle inequality; the mixed higher derivatives are the canonical-order weak derivatives $D^{β}$ of $C^{k}$ maps and multi-index derivative notation in Euclidean space.

[F4]

For $n \geq 1$ , $k \in N_{0}$ and $1 \leq p < \infty$ , the compactly supported smooth functions are dense in $W^{k, p} (R^{n})$ ; and on any open set, $C^{\infty} \cap W^{k, p}$ is dense in $W^{k, p}$ . (Compactly supported smooth functions are dense in W^{k,p}(R^n), Meyers–Serrin density on an arbitrary open set)

[F5]

For $g \in C^{2} ([a, b])$ one has $g (b) - g (a) = \int_{a}^{b} g^{'} (t) d t$ , and the weighted identity $\frac{1}{2 s} \int_{- s}^{s} (g^{'} (t) - g^{'} (0)) d t = \frac{1}{2 s} \int_{- s}^{s} sign (t) (s - ∣ t ∣) g^{''} (t) d t$ for $g \in C^{2} ([- s, s])$ ; iterated integrals of continuous functions over a triangle may be exchanged, and Lebesgue measure is invariant under translations of $R^{n}$ . (Botsko's theorem: if $F$ is continuous on $[a, b]$ , $F^{'} (x) = f (x)$ off a countable subset of $(a, b)$ , and $f$ is Riemann integrable, then $\int_{a}^{b} f = F (b) - F (a)$ , A region between two continuous graphs is Jordan measurable, and a continuous integrand extending to its closure integrates by vertical sections, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation)

[F6]

For a nonnegative measurable $f$ on a finite-measure set $E$ , $(\int_{E} f)^{p} \leq ∣ E ∣^{p - 1} \int_{E} f^{p}$ ; this is Hölder with the pair $(p, p / (p - 1))$ and the constant function. (Holder's inequality for integrals, including the endpoint cases)

Proof

technique · direct

1.1F1F2F3givenalgebra

The multiplier symbol. Fix an index $i$ and, for $ε > 0$ , define $m_{ε} (ξ) : = \frac{2 π i ξ_{i}}{4 π^{2} ε ∣ ξ ∣^{2} + 1}$ for $ξ \in R^{n}$ . Writing $η : = 2 π \sqrt{ε} ξ$ and $g (η) : = η_{i} / (∣ η ∣^{2} + 1)$ , we have $m_{ε} (ξ) = i ε^{- 1 / 2} g (η)$ and hence $\partial_{ξ}^{α} m_{ε} (ξ) = i ε^{- 1 / 2} (2 π \sqrt{ε})^{∣ α ∣} (\partial^{α} g) (η)$ ; therefore $∣ ξ ∣^{∣ α ∣} ∣ \partial_{ξ}^{α} m_{ε} (ξ) ∣ = ε^{- 1 / 2} ∣ η ∣^{∣ α ∣} ∣ (\partial^{α} g) (η) ∣ \leq C_{α} ε^{- 1 / 2}$ and $∥ m_{ε} ∥_{\infty} = ε^{- 1 / 2} \sup_{η} ∣ η ∣ / (1 + ∣ η ∣^{2}) = \frac{1}{2} ε^{- 1 / 2}$ , because $g$ and all of its derivatives are bounded on $R^{n}$ and $∣ η ∣^{∣ α ∣} ∣ \partial^{α} g ∣$ is bounded (near zero the derivatives of $g$ are bounded, while for $∣ η ∣ \geq 1$ the quotient rule gives $∣ \partial^{α} g (η) ∣ \leq C_{α} ∣ η ∣^{- 1 - ∣ α ∣}$ ). Thus $m_{ε}$ is a Mihlin symbol with constants $C_{α} ε^{- 1 / 2}$ , and [F1] gives the $L^{p}$ bound $∥ T_{m_{ε}} h ∥_{L^{p}} \leq C_{n, p} ε^{- 1 / 2} ∥ h ∥_{L^{p}}$ for every $h \in L^{p}$ , and $T_{m_{ε}} (ε (- Δ u) + u) = \partial_{i} u$ for $u \in C_{c}^{\infty} (R^{n})$ .

1.2F5algebra

One-dimensional identity along a coordinate line. Let $v \in C^{2} (R^{n})$ , $y \in R^{n}$ , $s > 0$ and $i \in {1, \dots, n}$ , and put $g (t) : = v (y + t e_{i})$ for $t \in [- s, s]$ . Since $g^{'} (0) = \partial_{i} v (y)$ , $g^{'} (t) = \partial_{i} v (y + t e_{i})$ and $g^{''} (t) = \partial_{i}^{2} v (y + t e_{i})$ , the first identity of [F5] applied on $[- s, s]$ gives $\frac{1}{2 s} \int_{- s}^{s} g^{'} (t) d t = \frac{g (s) - g (- s)}{2 s}$ , while the weighted second identity of [F5] gives $∣ g^{'} (0) - \frac{g (s) - g (- s)}{2 s} ∣ = ∣ \frac{1}{2 s} \int_{- s}^{s} sign (t) (s - ∣ t ∣) g^{''} (t) d t ∣ \leq \frac{1}{2} \int_{- s}^{s} ∣ g^{''} (t) ∣ d t$ . Hence $∣ \partial_{i} v (y) ∣ \leq \frac{1}{2 s} ∣ v (y + s e_{i}) - v (y - s e_{i}) ∣ + \frac{1}{2} \int_{- s}^{s} ∣ \partial_{i}^{2} v (y + t e_{i}) ∣ d t$ .

2.1step 1.1F2F3algebra

Global form for smooth compact data. Let $u \in C_{c}^{\infty} (R^{n})$ . On the Fourier side $F (ε (- Δ u) + u) = (4 π^{2} ε ∣ ξ ∣^{2} + 1) \hat{u}$ by [F2], so $m_{ε} \cdot F (ε (- Δ u) + u) = 2 π i ξ_{i} \hat{u} = F (\partial_{i} u)$ ; two tempered distributions with the same Fourier transform are equal by [F2], hence $T_{m_{ε}} (ε (- Δ u) + u) = \partial_{i} u$ . Step 1.1 and $∥ Δ u ∥_{L^{p}} \leq n ∥ D^{2} u ∥_{L^{p}}$ therefore give $∥ \partial_{i} u ∥_{L^{p}} \leq C_{n, p} ε^{- 1 / 2} (ε n ∥ D^{2} u ∥_{L^{p}} + ∥ u ∥_{L^{p}})$ , that is $∥ \partial_{i} u ∥_{L^{p}} \leq C_{n, p} n \sqrt{ε} ∥ D^{2} u ∥_{L^{p}} + C_{n, p} ε^{- 1 / 2} ∥ u ∥_{L^{p}}$ ; replacing $\sqrt{ε} C_{n, p} n$ by a new $ε > 0$ yields the global form $∥ D u ∥_{L^{p}} \leq ε ∥ D^{2} u ∥_{L^{p}} + C_{n, p} ε^{- 1} ∥ u ∥_{L^{p}}$ for every smooth compactly supported $u$ and every $ε > 0$ .

2.2step 1.2F3F5F6algebra

Local estimate for smooth functions. Fix a ball $B_{R} (x_{0})$ , let $0 < s \leq R$ and $v \in C^{\infty} (B_{2 R} (x_{0}))$ ; the points $y \pm s e_{i}$ and $y + t e_{i}$ , $∣ t ∣ \leq s$ , lie in $B_{2 R} (x_{0})$ whenever $y \in B_{R} (x_{0})$ . Raise the inequality of step 1.2 to the power $p$ , use $(a + b)^{p} \leq 2^{p} (a^{p} + b^{p})$ , integrate over $y \in B_{R} (x_{0})$ and apply [F6] to the inner integral over $t \in [- s, s]$ : $\int_{B_{R}} ∣ \partial_{i} v ∣^{p} d y \leq 2^{2 p} [(2 s)^{- p} \int_{B_{R}} (∣ v (y + s e_{i}) ∣^{p} + ∣ v (y - s e_{i}) ∣^{p}) d y + 2^{- p} (2 s)^{p - 1} \int_{B_{R}} \int_{- s}^{s} ∣ \partial_{i}^{2} v (y + t e_{i}) ∣^{p} d t d y] .$ By translation invariance [F5] the first integral is at most $2 ∥ v ∥_{L^{p} (B_{2 R})}^{p}$ and the second at most $2 s ∥ D^{2} v ∥_{L^{p} (B_{2 R})}^{p}$ (the inner $y$ -integrals are integrals over translate balls contained in $B_{2 R} (x_{0})$ ). Taking the $p$ -th root and the maximum over $i$ gives $∥ D v ∥_{L^{p} (B_{R})} \leq C_{n, p} (s^{- 1} ∥ v ∥_{L^{p} (B_{2 R})} + s ∥ D^{2} v ∥_{L^{p} (B_{2 R})})$ for every $0 < s \leq R$ .

3.1step 2.1F3F4algebra

Density. Let $u \in W^{2, p} (R^{n})$ and let $u_{k} \in C_{c}^{\infty} (R^{n})$ satisfy $u_{k} \to u$ in $W^{2, p} (R^{n})$ , which exists by [F4]. Applying step 2.1 to $u_{k} - u_{l}$ and to $u_{k}$ and letting $k \to \infty$ gives, in the limit, $∥ D u ∥_{L^{p}} \leq ε ∥ D^{2} u ∥_{L^{p}} + C_{n, p} ε^{- 1} ∥ u ∥_{L^{p}}$ : all three norms converge along the sequence and the constant is unchanged. This is the global form of the statement for every $W^{2, p}$ class.

3.2step 2.2F3F4algebra

Choosing the scale and passing to $W^{2, p}$ on the ball. In step 2.2 put $s : = \min {R, ε R / (C_{n, p})}$ ; then $s \leq R$ and $s^{- 1} \leq \max {R^{- 1}, C_{n, p} ε^{- 1} R^{- 1}} \leq C_{n, p} ε^{- 1} R^{- 1}$ for $0 < ε \leq C_{n, p}$ , while for $ε > C_{n, p}$ the inequality is implied by the case $ε = C_{n, p}$ (the right-hand side is increasing in $ε$ ); hence for every $ε > 0$ there is $C (n, p, ε) < \infty$ with $∥ D v ∥_{L^{p} (B_{R})} \leq ε R ∥ D^{2} v ∥_{L^{p} (B_{2 R})} + C R^{- 1} ∥ v ∥_{L^{p} (B_{2 R})}$ for every smooth $v$ on $B_{2 R} (x_{0})$ . Finally let $u \in W^{2, p} (B_{2 R} (x_{0}))$ and approximate it in $W^{2, p} (B_{2 R} (x_{0}))$ by smooth functions on that ball, which exist by [F4]; the estimate is stable under this convergence, so it holds for $u$ as well. This is the scaled form of the statement, uniformly in $x_{0}$ and $R$ .

4.1step 3.1step 3.2A1given∎

Conclusion. The global form is step 3.1 and the scaled form is step 3.2. Both were derived using only the Countable Choice instances recorded in [A1], namely those of the Fourier, multiplier, Sobolev-density and measure-translation interfaces; no extension operator and no full Axiom of Choice is used, which is why the scaled form is stated with the doubled ball on the right. The strict range $1 < p < \infty$ is used in the Mihlin theorem and nowhere else; the first-order identity of step 1.2 and the absorption of step 2.2 are elementary.

Remarks

The scale choice in step 3.2 is the only place where the parameter $s$ is optimized; the equality of the two forms after renaming $ε$ in step 2.1 is the classical "absorb the intermediate norm" step of the Gagliardo–Nirenberg interpolation.
The undoubled ball form $∥ D u ∥_{L^{p} (B_{R})} \leq ε R ∥ D^{2} u ∥_{L^{p} (B_{R})} + C R^{- 1} ∥ u ∥_{L^{p} (B_{R})}$ is a stronger statement on a bounded domain; an extension theorem gives one proof, but no necessity of a choice axiom is asserted. The doubled form above is what the local $W^{2, p}$ estimates actually consume.

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The cutoff commutator in the local $W^{2, p}$ estimates

Statement

Assume Countable Choice. Let $n \geq 1$ , $1 0$ , and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ have bounded coefficients on $B_{R} (x_{0})$ with $∣ A ∣ \leq Λ$ , $∣ b ∣ \leq M_{b}$ and $∣ c ∣ \leq M_{c}$ , where $∣ A ∣ : = \max_{i, j} ∣ a^{i j} ∣$ . Let $η \in C_{c}^{\infty} (B_{R} (x_{0}))$ satisfy $∣ D η ∣ \leq C_{1} R^{- 1}$ and $∣ D^{2} η ∣ \leq C_{2} R^{- 2}$ , and let $u \in W^{2, p} (B_{R} (x_{0}))$ . Then almost everywhere $L (η u) = η L u + (a^{i j} + a^{j i}) (\partial_{i} η) \partial_{j} u + (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) u,$ and consequently $∥ L (η u) - η L u ∥_{L^{p}} \leq C_{n} (Λ C_{1} R^{- 1} ∥ D u ∥_{L^{p}} + (Λ C_{2} R^{- 2} + M_{b} C_{1} R^{- 1}) ∥ u ∥_{L^{p}}) .$ There is no second derivative of $u$ in the commutator. No symmetry assumption on the principal coefficient matrix $A = (a^{i j})$ is needed.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , $1 0$ , coefficients $a^{i j}, b^{i}, c \in L^{\infty} (B_{R} (x_{0}))$ with $∣ A ∣ \leq Λ$ , $∣ b ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ , a cutoff $η$ as in the statement, and $u \in W^{2, p} (B_{R} (x_{0}))$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the Sobolev interfaces below. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

A function of class $W^{2, p}$ has weak derivatives $D_{i} u$ and $D_{i} D_{j} u$ in $L^{p}$ , and the Sobolev norm is the $p$ -sum of their $L^{p}$ norms; the $L^{p}$ norm obeys the triangle inequality. (Integer-order Sobolev spaces and their norms)

[F2]

The operator is $L u = \sum_{i, j} a^{i j} \partial_{i} \partial_{j} u + \sum_{i} b^{i} \partial_{i} u + c u$ , with the stated entrywise bounds $∣ a^{i j} ∣ \leq Λ$ , $∣ b^{i} ∣ \leq M_{b}$ , and $∣ c ∣ \leq M_{c}$ almost everywhere. No symmetry of $A = (a^{i j})$ is assumed; in particular $∣ a^{i j} + a^{j i} ∣ \leq 2 Λ$ .

[F3]

On a set of finite measure, the $L^{p}$ norm of a product is at most the sup-norm of one factor times the $L^{p}$ norm of the other. (Holder's inequality for integrals, including the endpoint cases)

Proof

technique · direct

1.1F1givenalgebra

Product rule for the cutoff. Since $η \in C_{c}^{\infty} (B_{R} (x_{0}))$ , for every test function $φ \in C_{c}^{\infty} (B_{R} (x_{0}))$ the product $η φ$ is again a test function; the defining identity for the weak derivative of $u$ therefore gives $\int (η u) \partial_{i} φ = \int u \partial_{i} (η φ) - \int u (\partial_{i} η) φ = - \int (η D_{i} u + u \partial_{i} η) φ$ , where the classical product rule was used on $η φ$ . Hence $η u \in W^{1, p} (B_{R} (x_{0}))$ with $D_{i} (η u) = η D_{i} u + u \partial_{i} η$ almost everywhere, both terms lying in $L^{p}$ . Applying the same argument to the $W^{1, p}$ function $D_{j} u$ in place of $u$ gives $D_{i} (η D_{j} u) = η D_{i} D_{j} u + (\partial_{i} η) D_{j} u$ , and combining the two identities yields $D_{j} D_{i} (η u) = η D_{j} D_{i} u + (\partial_{j} η) D_{i} u + (\partial_{i} η) D_{j} u + u \partial_{i} \partial_{j} η$ almost everywhere, all terms in $L^{p}$ . This direct weak-derivative argument does not require $u$ or $D_{j} u$ to be bounded.

2.1step 1.1F2algebra

Expanding the operator. Multiplying the pointwise equation $L u = a^{i j} \partial_{i} \partial_{j} u + b^{i} \partial_{i} u + c u$ by $η$ and substituting the product rule of step 1.1 gives, almost everywhere, $L (η u) = a^{i j} (η \partial_{i} \partial_{j} u + \partial_{j} η \partial_{i} u + \partial_{i} η \partial_{j} u + \partial_{i} \partial_{j} η u) + b^{i} (η \partial_{i} u + \partial_{i} η u) + c η u = η L u + (a^{i j} + a^{j i}) (\partial_{i} η) \partial_{j} u + (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) u,$ where relabelling $i$ and $j$ in the first cross term gives its coefficient $a^{j i}$ ; no symmetry assumption is needed.

3.1step 2.1F1F2F3algebra

$L^{p}$ bound. By [F2] and the cutoff bounds, $∣ (a^{i j} + a^{j i}) \partial_{i} η \partial_{j} u ∣ \leq 2 Λ n^{2} C_{1} R^{- 1} ∣ D u ∣$ and $∣ (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) u ∣ \leq (Λ n^{2} C_{2} R^{- 2} + M_{b} n C_{1} R^{- 1}) ∣ u ∣$ almost everywhere. Taking $L^{p}$ norms, using the triangle inequality of [F1] and the multiplicativity of the norm against bounded factors [F3] gives $∥ L (η u) - η L u ∥_{L^{p}} \leq C_{n} (Λ C_{1} R^{- 1} ∥ D u ∥_{L^{p}} + (Λ C_{2} R^{- 2} + M_{b} C_{1} R^{- 1}) ∥ u ∥_{L^{p}})$ . The dimension constant absorbs the factor $2 n^{2}$ from the nonsymmetric cross coefficient.

4.1step 2.1step 3.1A1given∎

Conclusion. The identity of step 2.1 involves only $u$ , $D u$ and the coefficient fields, never $D^{2} u$ , and the bound of step 3.1 is exactly the commutator estimate of the statement; the constants depend only on $n$ and the coefficient bounds, not on $u, η$ beyond the stated cutoff constants, and no choice beyond [A1] is used.

Remarks

The two first-order terms do not cancel: they are the symmetric pair produced by the product rule, and they are the reason a local $W^{2, p}$ estimate needs the interpolation inequality to absorb $R^{- 1} ∥ D u ∥_{L^{p}}$ .
The cutoff is compactly supported in the ball, so $η u$ extends by zero to a $W^{2, p}$ function on $R^{n}$ ; this is the localization used in the interior estimates below.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Local $W^{2, p}$ regularity of weak solutions of the Poisson equation

Statement

Assume Countable Choice. Let $n \geq 2$ , $1 < p < \infty$ , let $B \subseteq R^{n}$ be a ball and let $u \in W^{1, 2} (B)$ satisfy $- Δ u = f$ in the distributional sense on $B$ with $f \in L^{p} (B)$ . Then $u \in W_{l o c}^{2, p} (B)$ , and for every open $B^{'} ⋐ B$ there is $C = C (n, p, B^{'}, B) < \infty$ with $∥ u ∥_{W^{2, p} (B^{'})} \leq C (∥ f ∥_{L^{p} (B)} + ∥ u ∥_{L^{2} (B)}) .$ No boundary regularity is asserted, and no decay of $u$ at infinity is assumed; the estimate is local in the interior only.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 2$ , $1 < p < \infty$ , a ball $B$ , a function $u \in W^{1, 2} (B)$ with $- Δ u = f$ in $D^{'} (B)$ and $f \in L^{p} (B)$ , and an open $B^{'} ⋐ B$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the choice-qualified Newtonian-potential, Riesz-transform, Fourier and Sobolev interfaces. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

For $f \in L_{c}^{1} (R^{n})$ the Newtonian potential $N f$ is locally integrable and $- Δ T_{N f} = T_{f}$ in $D^{'} (R^{n})$ ; it is smooth and harmonic off the support of $f$ . For compactly supported $f \in L^{p}$ the local Young bound $∥ N f ∥_{L^{p} (B_{R})} \leq ∥ Φ ∥_{L^{1} (B_{R + ρ})} ∥ f ∥_{L^{p}}$ holds when $supp f \subseteq B_{ρ}$ , and similarly for the first derivatives with $\nabla Φ \in L_{l o c}^{1}$ . (Newtonian potential of compactly supported data, Newtonian potentials solve the distributional Poisson equation, Young's convolution inequality under Countable Choice, Fundamental solution for the positive operator minus Laplacian)

[F3]

The Riesz transforms have $L^{2}$ norm at most $1$ , satisfy $\sum_{j} R_{j}^{2} = - i d$ , and extend boundedly to $L^{p}$ with norm at most $C_{n, p}$ ; their composition $R_{i} R_{j}$ has symbol $- ξ_{i} ξ_{j} / ∣ ξ ∣^{2}$ . The Fourier transform satisfies $F (\partial^{α} g) = (2 π i ξ)^{α} F g$ and is injective on tempered distributions; two locally integrable functions equal as distributions are equal almost everywhere; distributional differentiation is continuous for the distribution topology. (Riesz transforms on Euclidean space, Riesz transforms are L2 contractions and square to minus the identity in sum, The Riesz transforms are bounded on Lp, Fourier differentiation and multiplication identities on tempered distributions, Fourier transform is a topological automorphism of tempered distributions, Locally integrable functions embed in distributions, Distributional differentiation is continuous and commutes)

[F4]

A locally integrable weakly harmonic function on an open set is $C^{\infty}$ there, and for every compact $K$ contained in the open set and every multi-index $β$ with $∣ β ∣ \leq 2$ one has $\sup_{K} ∣ D^{β} h ∣ \leq C (K, Ω) ∥ h ∥_{L^{1} (Ω)}$ . (Locally integrable weakly harmonic functions are smooth, Interior derivative estimates for harmonic functions)

[F5]

Smooth cutoffs between concentric balls exist: for $B^{'} ⋐ B^{''} ⋐ B$ there is $η \in C_{c}^{\infty} (B)$ with $0 \leq η \leq 1$ and $η = 1$ on a neighbourhood of $\overline{B^{''}}$ . Meyers–Serrin supplies smooth $W^{k, p}$ approximation, without claiming compact support on an arbitrary open set. For compactly supported $L^{p}$ data, apply its $k = 0$ case on $R^{n}$ and multiply by a fixed smooth cutoff equal to one on the support; the approximants then have one common compact support. (A smooth bump between concentric Euclidean balls, Meyers–Serrin density on an arbitrary open set, Integer-order Sobolev spaces and their norms)

Proof

technique · direct

1.1F1F5givenA1

Localization. Choose a ball $B^{''}$ with $B^{'} ⋐ B^{''} ⋐ B$ and, by [F5], a cutoff $η \in C_{c}^{\infty} (B)$ with $η = 1$ on a neighbourhood of $\overline{B^{''}}$ ; put $g : = η f$ , extended by zero to $R^{n}$ , so that $g \in L_{c}^{p} (R^{n})$ and $g = f$ on $B^{''}$ . Let $w : = N (g)$ be the Newtonian potential of $g$ .

2.1F1F3F5step 1.1algebra

Hessian bound for smooth data without dividing by the frequency variable. Let $g \in C_{c}^{\infty}$ and $w = N g$ . The classical-potential supplier Hölder data give a classical Newtonian solution gives $w \in C^{2}$ and $- Δ w = g$ . Fix $χ \in C_{c}^{\infty} (B_{2})$ equal to one on $B_{1}$ , and put $χ_{T} (x) = χ (x / T)$ . For large $T$ containing the support of $g$ , $Δ (χ_{T} w) = - g + 2 \nabla χ_{T} \cdot \nabla w + w Δ χ_{T}$ . On $T \leq ∣ x ∣ \leq 2 T$ , the kernel formulas and differentiation away from the support give $∣ w (x) ∣ \leq C_{g} T^{2 - n}$ for $n \geq 3$ , $∣ w (x) ∣ \leq C_{g} (1 + \log T)$ for $n = 2$ , and $∣ \nabla w (x) ∣ \leq C_{g} T^{1 - n}$ in both cases. Thus the commutator has $L^{p}$ norm at most $C_{g} T^{- n + n / p} (1 + 1_{n = 2} \log T)$ , which tends to zero for $p > 1$ . The whole-space estimate Global $W^{2, p}$ estimate for the Laplacian on Euclidean space applies to the compactly supported $C^{2}$ function $χ_{T} w$ (its classical derivatives are weak derivatives by integration by parts). On any fixed ball $B_{M}$ , $χ_{T} w = w$ for $T > M$ , so $∥ D^{2} w ∥_{L^{p} (B_{M})} \leq C_{n, p} (∥ g ∥_{p} + o (1))$ . First let $T \to \infty$ , then $M \to \infty$ ; Monotone convergence for the integral applied to the increasing ball indicators times the nonnegative Hessian integrands gives $∥ D^{2} w ∥_{L^{p} (R^{n})} \leq C_{n, p} ∥ g ∥_{p}$ . This includes $p = 2$ and avoids any two-dimensional Fourier inversion at zero.

3.1step 1.1step 2.1F1F3F5algebra

Second derivatives of $w$ : the $L^{p}$ case. For general $g \in L_{c}^{p}$ , choose $g_{k} \in C_{c}^{\infty}$ with $g_{k} \to g$ in $L^{p}$ (possible by [F5] after multiplying by a cutoff). By [F1] the potentials $N g_{k}$ converge to $N g$ in $L_{l o c}^{1}$ , and by the $L^{p}$ bound of step 2.1 the fields $D_{i j} N g_{k}$ are Cauchy in $L^{p} (R^{n})$ (apply step 2.1 to $g_{k} - g_{ℓ}$ ). Completeness of scalar $L^{p}$ follows from Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ for real components and Complex Lp completeness and almost-everywhere subsequences for complex data under Countable Choice. Since distributional differentiation is continuous [F3], the limit is $D_{i j} N g \in L^{p}$ , so $w = N g \in W_{l o c}^{2, p} (R^{n})$ with, if $supp g \subseteq B_{ρ} (0)$ , for every ball $B_{R} (0)$ , $∥ w ∥_{W^{2, p} (B_{R})} \leq C (n, p, R, ρ) ∥ g ∥_{L^{p}}$ (the zero- and first-order terms are controlled by the Young bounds of [F1] and the second-order terms by step 2.1).

4.1step 1.1step 3.1F1F4algebra

The remainder is harmonic. Since $η = 1$ on $B^{''}$ we have $g = f$ there, so $- Δ (u - w) = f - g = 0$ in $D^{'} (B^{''})$ by [F1]; hence $h : = u - w$ is a weakly harmonic function on $B^{''}$ and therefore $C^{\infty}$ there by [F4]. The interior derivative estimates give $∥ h ∥_{W^{2, p} (B^{'})} \leq C (n, B^{'}, B^{''}) ∥ h ∥_{L^{1} (B^{''})}$ , and $∥ h ∥_{L^{1} (B^{''})} \leq ∣ B ∣^{1 / 2} ∥ u ∥_{L^{2} (B)} + ∥ w ∥_{L^{1} (B^{''})} \leq C (B, n, p) (∥ u ∥_{L^{2} (B)} + ∥ f ∥_{L^{p} (B)})$ by the local Young bound [F1], Hölder on the bounded supports, and $∥ u ∥_{L^{1} (B^{''})} \leq ∣ B ∣^{1 / 2} ∥ u ∥_{L^{2} (B)}$ .

5.1step 3.1step 4.1F1F5given∎

Conclusion. On $B^{'}$ one has $u = w + h$ with $w \in W^{2, p} (B^{'})$ by step 3.1 and $h \in W^{2, p} (B^{'})$ by step 4.1, so $u \in W^{2, p} (B^{'})$ and $∥ u ∥_{W^{2, p} (B^{'})} \leq ∥ w ∥_{W^{2, p} (B^{'})} + ∥ h ∥_{W^{2, p} (B^{'})} \leq C (n, p, B^{'}, B) (∥ f ∥_{L^{p} (B)} + ∥ u ∥_{L^{2} (B)})$ , using $∥ g ∥_{L^{p}} \leq ∥ η ∥_{\infty} ∥ f ∥_{L^{p} (B)}$ . No boundary condition on $u$ was used, and the constants depend only on $n, p$ and the balls.

Remarks

The proof isolates the two inputs: the growing-cutoff whole-space estimate bounds the Hessian of the potential on $L^{p}$ data, while the harmonic remainder is controlled by the interior estimates for harmonic functions. The harmonic remainder is estimated in local $L^{1}$ ; no $L^{p}$ to $L^{2}$ embedding for the potential is assumed.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Interior $W^{2, p}$ estimate for uniformly elliptic equations with continuous coefficients

Statement

Assume Countable Choice. Let $n \geq 2$ , $1 0$ , $x_{0} \in R^{n}$ , and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic on $B_{R} (x_{0})$ with constants $λ, Λ$ , continuous principal coefficients on the closed ball, and $∥ b ∥_{\infty} + ∥ c ∥_{\infty} \leq M$ . Then every $u \in W^{2, p} (B_{R} (x_{0}))$ with $L u = f \in L^{p}$ satisfies the scale-invariant estimate $\sum_{j = 0}^{2} R^{j - 2} \max_{∣ β ∣ = j} ∥ D^{β} u ∥_{L^{p} (B_{R / 2} (x_{0}))} \leq C (R^{- 2} ∥ u ∥_{L^{p} (B_{R} (x_{0}))} + ∥ f ∥_{L^{p} (B_{R} (x_{0}))}),$ where $D^{0} u = u$ and the maximum runs over multi-indices of order $j$ , where $C$ may depend on $n, p, λ, Λ, R ∥ b ∥_{\infty}, R^{2} ∥ c ∥_{\infty}$ and the modulus of continuity of $A$ on the ball. A radius-independent constant requires uniform control of these dimensionless lower-order bounds and of the modulus. The theorem assumes continuity of $A$ ; no estimate for merely measurable principal coefficients is asserted.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 2$ , $1 0$ , $x_{0}$ , an operator $L$ with continuous uniformly elliptic principal part on ${\overset{ˉ}{B}}_{R} (x_{0})$ and $∥ b ∥_{\infty} + ∥ c ∥_{\infty} \leq M$ , and $u \in W^{2, p} (B_{R} (x_{0}))$ with $L u = f \in L^{p} (B_{R} (x_{0}))$ .

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the Sobolev, Fourier and multiplier interfaces. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The global estimate for the Laplacian: for $1 < p < \infty$ and $w \in W^{2, p} (R^{n})$ , $∥ D^{2} w ∥_{L^{p}} \leq C_{n, p} ∥ Δ w ∥_{L^{p}}$ ; the norm is the max over the second derivatives, equivalent to the Sobolev sum norm. (Global $W^{2, p}$ estimate for the Laplacian on Euclidean space, Integer-order Sobolev spaces and their norms)

[F2]

If $A_{0}$ is a symmetric positive-definite matrix with spectrum in $[λ, Λ]$ and $w \in W^{2, p} (R^{n})$ , put $S = A_{0}^{1 / 2}$ , $Φ (y) = x_{0} + S y$ , and $v = w \circ Φ$ . Testing the weak-derivative identities and changing variables by $Φ$ gives $D_{y_{i}} v = \sum_{k} S_{k i} (D_{x_{k}} w) \circ Φ$ and $D_{y_{i} y_{j}}^{2} v = \sum_{k, ℓ} S_{k i} S_{ℓ j} (D_{x_{k} x_{ℓ}}^{2} w) \circ Φ$ as $L^{p}$ classes; thus $v \in W^{2, p} (R^{n})$ . The change-of-variables formula gives $∥ g \circ Φ ∥_{L^{p} (d y)} = ∣ \det S ∣^{- 1 / p} ∥ g ∥_{L^{p} (d x)}$ , so the Hessian norms before and after pullback are equivalent with constants depending only on $n, p, λ, Λ$ , since $∥ S ∥ \leq \sqrt{Λ}$ and $∥ S^{- 1} ∥ \leq λ^{- 1 / 2}$ . Finally $Δ_{y} v = (A_{0} : D_{x}^{2} w) \circ Φ$ . Therefore the global Laplacian estimate [F1] gives $∥ D^{2} w ∥_{L^{p}} \leq C (n, p, λ, Λ) ∥ A_{0} : D^{2} w ∥_{L^{p}}$ . (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, Integer-order Sobolev spaces and their norms, Uniformly elliptic nondivergence-form operators and their frozen coefficients)

[F3]

Interpolation with $ε$ -loss on the whole space: $∥ D w ∥_{L^{p} (R^{n})} \leq ε R ∥ D^{2} w ∥_{L^{p} (R^{n})} + C (n, p, ε) R^{- 1} ∥ w ∥_{L^{p} (R^{n})}$ for every $w \in W^{2, p} (R^{n})$ and $R > 0$ . Its doubled-ball form also gives $∥ D u ∥_{L^{p} (B_{ρ})} \leq ε ρ ∥ D^{2} u ∥_{L^{p} (B_{2 ρ})} + C ρ^{- 1} ∥ u ∥_{L^{p} (B_{2 ρ})}$ . ( $L^{p}$ interpolation absorption of first derivatives by second derivatives)

[F4]

The cutoff identity: for $η \in C_{c}^{\infty} (B_{ρ}) \subset C_{c}^{\infty} (R^{n})$ and $w \in W^{2, p} (B_{ρ})$ , a.e. $L (η w) = η L w + 2 a^{i j} (\partial_{i} η) \partial_{j} w + (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) w$ , and $η w$ extends by zero to a $W^{2, p} (R^{n})$ function; moreover $η D^{2} w = D^{2} (η w) - D η \otimes D w - D w \otimes D η - w D^{2} η$ in the sense of $L^{p}$ classes on $supp η$ . (The cutoff commutator in the local $W^{2, p}$ estimates)

Proof

technique · direct

1.1F1F2F3F4givenA1algebra

Frozen estimate on nested balls. Fix $x \in B_{R} (x_{0})$ and $0 < ρ < R / 4$ with $B_{2 ρ} (x) \subset B_{R} (x_{0})$ . Freeze $A$ at $A_{0} = A (x)$ and choose $η \in C_{c}^{\infty} (B_{ρ} (x))$ with $0 \leq η \leq 1$ and $η = 1$ on $B_{ρ / 2} (x)$ , with $∣ D η ∣ \leq C_{n} ρ^{- 1}$ and $∣ D^{2} η ∣ \leq C_{n} ρ^{- 2}$ . Set $w = η u$ , extended by zero. The constant-coefficient estimate [F2] and the product identity [F4] give $∥ D^{2} u ∥_{L^{p} (B_{ρ / 2} (x))} \leq ∥ D^{2} w ∥_{L^{p} (R^{n})} \leq C ∥ A_{0} : D^{2} w ∥_{L^{p} (R^{n})} \leq C (∥ f ∥_{L^{p} (B_{ρ} (x))} + ω_{A} (ρ) ∥ D^{2} u ∥_{L^{p} (B_{ρ} (x))} + (M_{b} + Λ ρ^{- 1}) ∥ D u ∥_{L^{p} (B_{ρ} (x))} + (M_{c} + Λ ρ^{- 2}) ∥ u ∥_{L^{p} (B_{ρ} (x))}),$ where $ω_{A} (ρ) : = \sup {∣ A (y) - A (z) ∣ : y, z \in {\overset{ˉ}{B}}_{R} (x_{0}), ∣ y - z ∣ \leq ρ}$ and the constant $C$ depends only on $n, p, λ, Λ$ . Apply the doubled-ball interpolation inequality [F3] to $u$ on $B_{2 ρ} (x)$ : $∥ D u ∥_{L^{p} (B_{ρ} (x))} \leq ε ρ ∥ D^{2} u ∥_{L^{p} (B_{2 ρ} (x))} + C_{n, p, ε} ρ^{- 1} ∥ u ∥_{L^{p} (B_{2 ρ} (x))} .$ Since $∥ D^{2} u ∥_{L^{p} (B_{ρ})} \leq ∥ D^{2} u ∥_{L^{p} (B_{2 ρ})}$ , choose $ε > 0$ small and then $ρ_{*} > 0$ so that for every $0 < ρ \leq ρ_{*}$ the coefficient of $∥ D^{2} u ∥_{L^{p} (B_{2 ρ} (x))}$ after substitution is at most any prescribed $θ_{0} > 0$ ; this is possible because $ω_{A} (ρ) \to 0$ and $ρ M_{b} \leq R M_{b}$ . Absorbing constants in the lower-order term yields the local estimate $∥ D^{2} u ∥_{L^{p} (B_{ρ / 2} (x))} \leq C_{0} ∥ f ∥_{L^{p} (B_{ρ} (x))} + C_{0} ρ^{- 2} ∥ u ∥_{L^{p} (B_{2 ρ} (x))} + θ_{0} ∥ D^{2} u ∥_{L^{p} (B_{2 ρ} (x))},$ where $C_{0}$ depends on $n, p, λ, Λ, R M_{b}, R^{2} M_{c}$ and the modulus of continuity of $A$ , but not on $x$ or $ρ \leq ρ_{*}$ . The cutoff is identically one on the smaller ball, so the left side is the unweighted Hessian norm there; no division by a vanishing cutoff is used.

2.1step 1.1F1F3inductionalgebra∎

Finite-overlap cover and hole filling. Write $M (r) : = ∥ D^{2} u ∥_{L^{p} (B_{r} (x_{0}))}$ and $U : = ∥ u ∥_{L^{p} (B_{R} (x_{0}))}$ , $F : = ∥ f ∥_{L^{p} (B_{R} (x_{0}))}$ . For $r < R$ with $δ : = s - r > 0$ sufficiently small, put $ρ = δ / 4$ and cover $B_{r} (x_{0})$ by the balls $B_{ρ / 2} (x_{j})$ of a cubic lattice of mesh $ρ / (4 \sqrt{n})$ ; the enlarged balls $B_{2 ρ} (x_{j})$ lie in $B_{s} (x_{0})$ and have overlap bounded by a constant depending only on $n$ . Applying step 1.1 on each patch and taking the $p$ -sum, the finite-overlap bounds give $M (r) \leq C_{1} F + C_{1} δ^{- 2} U + θ M (s),$ where $θ$ can be fixed in advance as small as desired by choosing $θ_{0}$ small enough relative to the overlap constant, and $C_{1}$ is independent of $r, s$ (it may depend on $ρ_{*}^{- 1}$ only through the permitted modulus-of-continuity dependence). Choose $θ < 1 / 4$ . Take $δ_{j} = δ_{0} 2^{- j}$ with $0 < 2 δ_{0} \leq R / 4$ and $δ_{0} / 4 \leq ρ_{*}$ , and put $r_{0} = 3 R / 4$ , $r_{j + 1} = r_{j} + δ_{j}$ ; then $r_{j} ↑ r_{\infty} \leq R$ . Iterating gives $M (r_{0}) \leq C_{1} F \sum_{j = 0}^{N - 1} θ^{j} + C_{1} U \sum_{j = 0}^{N - 1} θ^{j} δ_{j}^{- 2} + θ^{N} M (r_{N}) .$ The first series is bounded, the second converges because $δ_{j}^{- 2} = δ_{0}^{- 2} 4^{j}$ and $4 θ < 1$ , and the final term tends to zero since $u \in W^{2, p} (B_{R} (x_{0}))$ . Therefore $∥ D^{2} u ∥_{L^{p} (B_{3 R / 4})} \leq C (F + R^{- 2} U)$ , with $δ_{0}^{- 2}$ written as $R^{- 2}$ times a constant depending on the permitted dimensionless radius ratio. To control first derivatives on $B_{R / 2}$ , choose a cubic lattice of mesh $R / (32 \sqrt{n})$ and retain the finitely many centers $x_{j} \in B_{9 R / 16} (x_{0})$ whose balls $B_{R / 16} (x_{j})$ meet $B_{R / 2} (x_{0})$ . These inner balls cover $B_{R / 2}$ : every point is within $R / 64$ of a lattice point, and such a point lies in $B_{9 R / 16}$ . Their doubled balls $B_{R / 8} (x_{j})$ lie in $B_{11 R / 16} (x_{0}) \subset B_{3 R / 4} (x_{0})$ . Apply the doubled-ball interpolation inequality [F3] with radius $R / 16$ on each patch and take the finite $p$ -sum; bounded overlap gives $R^{- 1} ∥ D u ∥_{L^{p} (B_{R / 2})} \leq C_{n} (∥ D^{2} u ∥_{L^{p} (B_{3 R / 4})} + R^{- 2} U) \leq C (F + R^{- 2} U) .$ The zeroth-order term satisfies $R^{- 2} ∥ u ∥_{L^{p} (B_{R / 2})} \leq R^{- 2} U$ . Combining this with the Hessian bound proves the displayed scale-invariant estimate. The exponent range is $1 < p < \infty$ as in [F1] and [F3], and the constant has exactly the stated dependence.

Remarks

The proof is the standard freezing argument: the frozen constant-coefficient operator is controlled by the global Laplacian estimate after a linear change of variables, and the coefficient oscillation on a small ball is absorbed with the interpolation inequality; the patching over the cover globalizes the local estimate to $B_{R / 2} (x_{0})$ .
Continuity of the principal coefficients is used only to make the oscillation $\sup_{B_{ρ}} ∣ A - A (x_{0}) ∣$ arbitrarily small by choosing $ρ$ ; no Hölder regularity is asserted or needed in this scale.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Global $W^{2, p}$ Dirichlet estimate on a $C^{1, 1}$ domain

Statement

Assume the Axiom of Choice and Countable Choice. Let $n \geq 2$ , $1 < p < \infty$ , let $Ω$ be a bounded $C^{1, 1}$ domain, and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic on $Ω$ with $a^{i j} \in C^{0} (\overset{ˉ}{Ω})$ , $∥ b ∥_{\infty} + ∥ c ∥_{\infty} \leq M$ , and ellipticity constants $λ, Λ$ . Then there is $C < \infty$ , depending on $n, p, λ, Λ, M, Ω$ and the modulus of continuity of $A$ , such that every $u \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ satisfies $∥ u ∥_{W^{2, p} (Ω)} \leq C (∥ L u ∥_{L^{p} (Ω)} + ∥ u ∥_{L^{p} (Ω)}) .$ The estimate is a priori and asserts neither solvability nor weak-to-strong regularity: the function is assumed to lie in $W^{2, p}$ with zero trace.

Facts & Assumptions

Given: the Axiom of Choice and ${A C}_{ω}$ , $n \geq 2$ , $1 < p < \infty$ , the bounded $C^{1, 1}$ domain $Ω$ , the operator $L$ with the stated bounds, and $u \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ .

[A1]

The Axiom of Choice is inherited by the half-space trace, extension and Lipschitz/Sobolev interfaces; Countable Choice is inherited by the estimate and measure interfaces. (The Axiom of Choice) All covers and bump families are finite and explicitly exhibited. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The spaces $W^{2, p}$ and $W_{0}^{1, p}$ are those of Integer-order Sobolev spaces and their norms and Zero-boundary Sobolev space as a norm closure; in particular, $W_{0}^{1, p} (Ω)$ is the $W^{1, p}$ closure of $C_{c}^{\infty} (Ω)$ , and multiplication by a smooth compactly supported cutoff preserves that closure, by multiplying the approximating test functions. (A smooth bump between concentric Euclidean balls)

[F2]

Interior $W^{2, p}$ estimate (Interior $W^{2, p}$ estimate for uniformly elliptic equations with continuous coefficients): under its stated hypotheses, every $w \in W^{2, p} (B_{R} (x_{0}))$ with $L w \in L^{p}$ satisfies the scale-invariant estimate $\sum_{j = 0}^{2} R^{j - 2} \max_{∣ β ∣ = j} ∥ D^{β} w ∥_{L^{p} (B_{R / 2} (x_{0}))} \leq C (R^{- 2} ∥ w ∥_{L^{p} (B_{R} (x_{0}))} + ∥ L w ∥_{L^{p} (B_{R} (x_{0}))}) .$ On any fixed patch radius this implies the corresponding unweighted $W^{2, p}$ estimate with a constant also depending on that radius, which is the form used below.

[F3]

Whole-space Laplace estimate (Global $W^{2, p}$ estimate for the Laplacian on Euclidean space): $∥ D^{2} w ∥_{L^{p} (R^{n})} \leq C_{n, p} ∥ Δ w ∥_{L^{p} (R^{n})}$ for every $w \in W^{2, p} (R^{n})$ and $1 < p < \infty$ , with the max-form convention for $∥ D^{2} w ∥_{L^{p}}$ up to dimensional constants.

[F4]

Half-space Dirichlet estimate for constant coefficients. Let $A_{0}$ be symmetric positive definite with $λ ∣ ξ ∣^{2} \leq A_{0}^{i j} ξ_{i} ξ_{j} \leq Λ ∣ ξ ∣^{2}$ , let $H = {x_{n} > 0}$ and let $v \in W^{2, p} (H) \cap W_{0}^{1, p} (H)$ . Then $∥ \partial_{i j} v ∥_{L^{p} (H)} \leq C_{n, p, λ, Λ} ∥ A_{0}^{i j} \partial_{i} \partial_{j} v ∥_{L^{p} (H)} .$ Proof: put $y : = A_{0}^{- 1 / 2} x$ , so $x = A_{0}^{1 / 2} y$ maps $H$ onto the half-space $H^{'} = {y \cdot ν > 0}$ with $ν : = A_{0}^{1 / 2} e_{n}$ , and define $\tilde{v} (y) : = v (A_{0}^{1 / 2} y)$ . The chain rule gives $Δ_{y} \tilde{v} (y) = (A_{0}^{i j} \partial_{i} \partial_{j} v) (A_{0}^{1 / 2} y)$ ; after a rotation $H^{'}$ is ${y_{n} > 0}$ and the Laplacian is invariant. Let $V$ be the odd extension of $\tilde{v}$ in the normal variable $y_{n}$ . Its trace is zero; The trace operator of The half-space trace estimate and the half-space trace operator applies to $\tilde{v}$ and its first derivatives. Approximate $\tilde{v}$ by functions smooth up to the face by applying Integer-order Sobolev extension from a half-space and whole-space smooth density. Trace continuity and tangential integration by parts against a compact boundary test give $T (\partial_{a} \tilde{v}) = \partial_{a} (T \tilde{v}) = 0$ for $a < n$ . The odd extension has zero function trace; its normal derivative is even, so its two traces agree, while each tangential derivative is odd with zero trace. Integration by parts on the two half-spaces therefore produces no interface distributions through order two, so $V \in W^{2, p} (R^{n})$ and $Δ V$ is the odd extension of $Δ \tilde{v}$ . Thus $∥ Δ V ∥_{L^{p} (R^{n})} = 2^{1 / p} ∥ Δ \tilde{v} ∥_{L^{p} (H^{'})}$ , with the same factor for each second derivative. Applying [F3] to $V$ and changing variables back through $A_{0}^{\pm 1 / 2}$ (whose operator norms are bounded by $\max {λ^{- 1 / 2}, Λ^{1 / 2}}$ , with Jacobian factors likewise controlled by $λ, Λ$ ) gives the claim. The reflection and weak derivative compatibility are established here; Wang's Schauder Theorem 1' is not an Lp supplier.

[F5]

Cutoff commutator (The cutoff commutator in the local $W^{2, p}$ estimates): for $η \in C_{c}^{\infty} (B_{R} (x_{0}))$ and $w \in W^{2, p} (B_{R} (x_{0}))$ , $L (η w) = η L w + (a^{i j} + a^{j i}) (\partial_{i} η) \partial_{j} w + (a^{i j} \partial_{i} \partial_{j} η + b^{i} \partial_{i} η) w a.e.$ with the corresponding $L^{p}$ commutator bound. In this theorem's symmetric principal-matrix convention, $a^{i j} = a^{j i}$ , so the cross term specializes to $2 a^{i j} (\partial_{i} η) \partial_{j} w$ and the previous bound with $2 Λ$ is valid. The identity applies on each flattened half-box with its transformed symmetric principal matrix.

[F6]

Absorption ( $L^{p}$ interpolation absorption of first derivatives by second derivatives): for every $ε > 0$ there is $C (n, p, ε)$ with $∥ D w ∥_{L^{p} (R^{n})} \leq ε ∥ D^{2} w ∥_{L^{p} (R^{n})} + C ∥ w ∥_{L^{p} (R^{n})}$ for $w \in W^{2, p} (R^{n})$ , and on balls $∥ D w ∥_{L^{p} (B_{R} (x_{0}))} \leq ε R ∥ D^{2} w ∥_{L^{p} (B_{2 R} (x_{0}))} + C R^{- 1} ∥ w ∥_{L^{p} (B_{2 R} (x_{0}))}$ .

[F7]

Local $C^{1, 1}$ flattening. Here $C^{1, 1}$ means that each local boundary graph is $C^{1}$ and its first derivatives are Lipschitz on compact patches. The flattening map is a shear with determinant one; its first derivative and inverse are bounded, and its second weak derivatives are essentially bounded: apply the real-valued converse of $W^{1, \infty}$ functions on convex domains have Lipschitz representatives to each Lipschitz graph-gradient component on a smaller convex base box. For smooth $w$ , mollify the graph on a slightly larger base box. The graph functions and gradients converge uniformly; their uniformly bounded second derivatives converge locally in each finite $L^{p}$ to the weak second derivatives. Apply the ordinary chain rule to the smooth shears, then pass to the weak derivative identities using change of variables and these convergences. This gives $D (w \circ Φ) = (D w \circ Φ) D Φ$ and $D^{2} (w \circ Φ) = (D^{2} w \circ Φ) [D Φ, D Φ] + (D w \circ Φ) D^{2} Φ$ . The change-of-variables formula and the bound $∥ g h ∥_{L^{p}} \leq ∥ g ∥_{L^{\infty}} ∥ h ∥_{L^{p}}$ therefore bound the local $W^{1, p}$ and $W^{2, p}$ norms of the pullback by the corresponding original norms, with constants controlled by the chart bounds. For general $w \in W^{2, p}$ , smooth approximation on the open chart patch (Meyers–Serrin density on an arbitrary open set) and weak stability of derivatives (Weak derivatives persist under local Lp limits) give the same weak chain rule and estimate. The Jacobian and chart derivatives are controlled on the finite atlas of the bounded $C^{1, 1}$ domain (Bounded C^k domains and boundary charts, A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Classical derivatives agree with weak derivatives, Holder's inequality for integrals, including the endpoint cases). The transformed lower-order coefficients are bounded by the original bounds and the chart's $C^{1, 1}$ bounds. (Uniformly elliptic nondivergence-form operators and their frozen coefficients)

Proof

technique · direct

1.1F1F7givenconstructA1

Uniform charts, transformed-coefficient modulus, and bounded overlap. Start with a finite $C^{1, 1}$ boundary atlas and a finite interior atlas. In a boundary chart $x = Φ (y)$ the transformed principal matrix is $\tilde{A} (y) = D Φ (y)^{- 1} A (Φ (y)) D Φ (y)^{- T}$ . The chart maps and their first derivatives are uniformly bounded on the finite atlas, and $D Φ$ is Lipschitz; therefore, for a modulus $ω_{A}$ of $A$ on $\overset{ˉ}{Ω}$ , $ω_{\tilde{A}} (s) \leq C_{0} (ω_{A} (C_{0} s) + s)$ on every chart, with one $C_{0}$ for the finite atlas. Choose a small scale $r$ so the half-space estimate constant times $ω_{\tilde{A}} (C_{0} r)$ is as small as required below. A grid in each chart and in the interior gives inner patches covering $Ω$ : boundary half-patches cover a collar of width comparable to $r$ , and the remaining interior balls have doubled balls contained in $Ω$ . Choose fixed larger patches for cutoffs and the doubled-ball interpolation inequality, still inside the chart or $Ω$ . The expanded patches have overlap at most $N_{0}$ , independent of $r$ ; smooth cutoffs equal to one on inner patches and supported on the expanded patches have first and second derivatives bounded by $C r^{- 1}$ and $C r^{- 2}$ . The grid overlap and finite atlas fix $N_{0}$ before we choose the local Hessian error.

2.1F2step 1.1algebra

Interior patches with doubled balls inside $Ω$ . For an interior member $B_{r} (x_{ν})$ , step 1.1 ensures $B_{2 r} (x_{ν}) \subset Ω$ . Apply [F2] to $u$ on $B_{2 r} (x_{ν})$ to obtain $∥ u ∥_{W^{2, p} (B_{r} (x_{ν}))} \leq C (r^{- 2} ∥ u ∥_{L^{p} (B_{2 r} (x_{ν}))} + ∥ L u ∥_{L^{p} (B_{2 r} (x_{ν}))}) .$ These doubled balls are all contained in $Ω$ , and their constants are uniform because the coefficient modulus and the dimensionless lower-order bounds are controlled at the fixed scale.

2.2F1F3F4F5F6F7step 1.1algebra

Boundary patches and a prescribed local error. In a boundary chart flatten the graph, write $\tilde{u} = u \circ Φ$ , let $\tilde{L}$ be the transformed operator, and take a cutoff $χ$ equal to one on an inner half-patch and supported on a larger half-patch. Set $v = χ \tilde{u}$ . To see $v \in W_{0}^{1, p} (R_{+}^{n})$ , choose $u_{k} \in C_{c}^{\infty} (Ω)$ converging to $u$ in $W^{1, p}$ by [F1]. Their pullbacks, multiplied by $χ$ , are compactly supported in the open half-space and lie in $W_{0}^{1, p} (R_{+}^{n})$ by mollification. The smooth W^{1,p} chart estimate [F7] makes these pullbacks Cauchy in $W^{1, p}$ ; the change-of-variables $L^{p}$ estimate identifies their limit with $v$ , so closedness of $W_{0}^{1, p}$ gives $v \in W_{0}^{1, p} (R_{+}^{n})$ . Since $u \in W^{2, p}$ , the same chart rule gives $v \in W^{2, p} (R_{+}^{n})$ ; extend it by zero away from the patch. Freeze the transformed principal matrix at the chart centre to get ${\tilde{L}}_{0}$ . The half-space estimate [F4], whose odd-reflection proof applies the whole-space estimate [F3], and the product identity [F5] bound $∥ D^{2} v ∥_{L^{p}}$ by the transformed $∥ L u ∥_{L^{p}}$ , the principal error $∥ χ ({\tilde{A}}_{0} - \tilde{A}) : D^{2} \tilde{u} ∥_{L^{p}}$ , lower-order products, and cutoff commutators bounded by $C (r^{- 1} ∥ D \tilde{u} ∥_{L^{p}} + r^{- 2} ∥ \tilde{u} ∥_{L^{p}})$ on the expanded patch. The identity $χ D^{2} \tilde{u} = D^{2} v - D χ \otimes D \tilde{u} - D \tilde{u} \otimes D χ - (D^{2} χ) \tilde{u}$ puts the principal error on the left with coefficient at most $C ω_{\tilde{A}} (C_{0} r)$ , which is made small in step 1.1. For the term $χ D \tilde{u}$ , write it as $D v - (D χ) \tilde{u}$ and apply the whole-space interpolation inequality [F6] to the odd extension of $v$ ; choose its parameter small enough to absorb the resulting $∥ D^{2} v ∥_{L^{p}}$ term. For the cutoff commutator, odd-extend $\tilde{u}$ across the flat face on the larger half-ball and apply the doubled-ball form of [F6], giving for every $ε > 0$ $r^{- 1} ∥ D \tilde{u} ∥_{L^{p} (P^{+})} \leq ε ∥ D^{2} \tilde{u} ∥_{L^{p} (P^{+ +})} + C_{ε} r^{- 2} ∥ \tilde{u} ∥_{L^{p} (P^{+ +})} .$ The chart chain rule [F7] also contributes bounded first-order terms when comparing second derivatives; the same interpolation absorbs them into an arbitrarily small multiple of the outer Hessian norm. Thus, after first fixing the transformed-coefficient oscillation and then choosing the interpolation parameters, for any prescribed $η > 0$ the boundary patch satisfies $∥ u ∥_{W^{2, p} (P)} \leq C_{1} (∥ L u ∥_{L^{p} (P^{+ +})} + r^{- 2} ∥ u ∥_{L^{p} (P^{+ +})}) + η ∥ D^{2} u ∥_{L^{p} (P^{+ +})},$ where $P$ is the inner patch and $P^{+}$ , $P^{+ +}$ are fixed expanded patches from step 1.1. All chart Jacobians and lower-order coefficient bounds enter $C_{1}$ , which is independent of $u$ .

3.1step 1.1step 2.1step 2.2algebra

Sum with bounded overlap and absorb quantitatively. The inner patches cover $Ω$ and the expanded patches have overlap at most $N_{0}$ . Taking the $ℓ^{p}$ sum of the local estimates in steps 2.1 and 2.2 therefore gives $∥ u ∥_{W^{2, p} (Ω)} \leq C_{2} (∥ L u ∥_{L^{p} (Ω)} + r^{- 2} ∥ u ∥_{L^{p} (Ω)}) + C_{o v} η ∥ D^{2} u ∥_{L^{p} (Ω)},$ where $C_{o v}$ depends only on the fixed overlap and chart constants. Now choose the local error from step 2.2 after this overlap constant is fixed so that $C_{o v} η \leq \frac{1}{2}$ . Since $∥ D^{2} u ∥_{L^{p}} \leq ∥ u ∥_{W^{2, p}}$ , the last term is absorbed into the left side. This yields the stated estimate; the finite-overlap factor is accounted for explicitly rather than assumed small.

4.1step 1.1step 3.1F4given∎

Conclusion. Step 3.1 yields the displayed a priori estimate with a constant depending on $n, p, λ, Λ, M, Ω$ , the finite $C^{1, 1}$ atlas and the modulus $ω_{A}$ through $ω_{\tilde{A}} (s) \leq C_{0} (ω_{A} (C_{0} s) + s)$ . The zero trace is used for the odd extension in the half-space estimate, and every interior doubled ball is contained in $Ω$ . The proof asserts neither solvability nor weak-to-strong regularity.

Remarks

The globalization has two ingredients: the interior estimate [F2] and the half-space Dirichlet estimate [F4], the latter used after flattening and freezing. The freezing radius is chosen once, uniformly over the finite atlas, using continuity of the principal coefficients on the compact set $\overset{ˉ}{Ω}$ ; this is where the modulus of continuity enters the constant.
The estimate is genuinely a priori: the odd reflection used in [F4] requires a function already in $W^{2, p}$ with zero trace. Obtaining that membership from a weak formulation is the content of the weak-to-strong regularity theorem of this page, not of this a priori estimate.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Global Schauder regularity for the weak Dirichlet Laplacian

Statement

Assume the Axiom of Choice together with Countable Choice. Let $n \geq 2$ , $0 < α < 1$ , let $Ω$ be a bounded $C^{2, α}$ domain, let $f \in C^{0, α} (\overset{ˉ}{Ω})$ and $g \in C^{2, α} (\overset{ˉ}{Ω})$ . Then the weak Dirichlet problem $- Δ u = f$ in $Ω$ , $u = g$ on $\partial Ω$ (understood as $u - g \in H_{0}^{1} (Ω)$ ) has exactly one solution $u \in H^{1} (Ω)$ , and this solution belongs to $C^{2, α} (\overset{ˉ}{Ω})$ , satisfies $- Δ u = f$ pointwise in $Ω$ and $u = g$ on $\partial Ω$ , and obeys $∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C (∥ f ∥_{C^{0, α} (\overset{ˉ}{Ω})} + ∥ g ∥_{C^{2, α} (\overset{ˉ}{Ω})})$ with $C = C (n, α, Ω)$ . This supplies the Laplace base point of the continuity method rather than assuming it.

Facts & Assumptions

Given: the Axiom of Choice and Countable Choice, $n \geq 2$ , $0 < α < 1$ , the bounded $C^{2, α}$ domain $Ω$ , and $f \in C^{0, α} (\overset{ˉ}{Ω})$ , $g \in C^{2, α} (\overset{ˉ}{Ω})$ .

[A1]

The proof assumes the Axiom of Choice and Countable Choice. Countable Choice is inherited by the measure and approximation interfaces; the Axiom of Choice is inherited by weak existence through its Poincaré supplier, and by the extension, trace, embedding, regularity and compactness interfaces. (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Weak formulation and solvability: the weak Dirichlet problem is the problem of $u \in H^{1} (Ω)$ with $u - g \in H_{0}^{1} (Ω)$ and $\int_{Ω} \nabla u \cdot \nabla φ = \int_{Ω} f φ$ for every $φ \in H_{0}^{1} (Ω)$ , where $g \in C^{2, α} (\overset{ˉ}{Ω})$ belongs to $H^{1} (Ω)$ and its trace $g ∣_{\partial Ω}$ lies in $H^{1 / 2} (\partial Ω)$ by The sharp trace theorem: boundedness and range in the fractional space with $p = 2$ ; The $L^{p}$ trace operator on a bounded $C^{1}$ domain identifies the trace with the classical boundary restriction. By The inhomogeneous weak Dirichlet problem by a trace lifting the problem has exactly one solution $u \in H^{1} (Ω)$ ; for the zero-boundary problem the solution is the one of Existence and uniqueness for the weak Dirichlet Poisson problem. The kernel of the trace is $H_{0}^{1} (Ω)$ (The kernel of the trace is the closure of the test functions), and for a continuous function on $\overset{ˉ}{Ω}$ the Sobolev trace equals its boundary restriction (The trace agrees with classical restriction for continuous Sobolev functions).

[F2]

Extension of H"older data: every $F \in C^{0, α} (\overset{ˉ}{Ω})$ extends to a compactly supported $F^{♯} \in C_{c}^{0, α} (R^{n})$ with $∥ F^{♯} ∥_{C^{0, α} (R^{n})} \leq C (Ω, α) ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})}$ : for real-valued $F$ take the McShane extension $F^{♯} (x) = \inf_{y \in \overset{ˉ}{Ω}} (F (y) + [F]_{0, α} ∣ x - y ∣^{α})$ and multiply by a fixed cutoff equal to $1$ on a neighbourhood of $\overset{ˉ}{Ω}$ ; complex-valued data are extended componentwise. The inequality $(a + b)^{α} \leq a^{α} + b^{α}$ gives $∣ ∣ x - y ∣^{α} - ∣ x^{'} - y ∣^{α} ∣ \leq ∣ x - x^{'} ∣^{α}$ ; taking infima proves the extension Hölder bound, and the original Hölder inequality makes the infimum equal to $F (x)$ when $x \in \overset{ˉ}{Ω}$ . Multiplication by the fixed smooth cutoff preserves the bound up to its fixed constant.

[F3]

Mollification smooths and controls: for a mollifier $ρ_{ε}$ , the convolutions $F_{ε} : = F^{♯} * ρ_{ε}$ lie in $C_{c}^{\infty} (R^{n})$ with $\sup ∣ F_{ε} ∣ \leq \sup ∣ F^{♯} ∣$ , $[F_{ε}]_{0, α} \leq [F^{♯}]_{0, α}$ and $F_{ε} \to F^{♯}$ uniformly on compact sets, in particular on $\overset{ˉ}{Ω}$ . (Convolution with a mollifier is smooth, and derivatives pass under the integral sign)

[F4]

Weak-to-strong global regularity (Weak global $W^{2, p}$ regularity for the Dirichlet Laplacian): if $p > n$ , $Ω$ is a bounded $C^{2, α}$ domain and $w \in H_{0}^{1} (Ω)$ is a weak solution of $- Δ w = h$ with $h \in L^{p} (Ω)$ , then $w \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ and $∥ w ∥_{W^{2, p}} \leq C (∥ h ∥_{L^{p}} + ∥ w ∥_{L^{p}})$ with $C = C (n, p, Ω)$ .

[F5]

Higher-order Sobolev embedding (Higher-order Sobolev embedding): on the bounded extension domain $Ω$ (Bounded C^k domains admit integer-order Sobolev extension), for $k \geq 1$ and $1 \leq q < \infty$ , if $k q < n$ then $W^{k, q} ↪ L^{r}$ for $r \leq n q / (n - k q)$ , if $k q = n$ then $W^{k, q} ↪ L^{r}$ for every finite $r$ , and if $k q > n$ then there are $C^{m, β}$ representatives for $m + β < k - n / q$ . In particular $W^{2, q_{i}} ↪ L^{q_{i + 1}}$ at each subcritical exponent in step 4.1, and for $p > n$ , $W^{2, p} (Ω)$ embeds in $C^{1, γ} (\overset{ˉ}{Ω})$ for every $0 < γ < 1 - n / p$ , with norm bounded by a constant times $∥ w ∥_{W^{2, p}}$ .

[F6]

Interior regularity for smooth forcing: if $h \in C_{c}^{\infty} (R^{n})$ then $N h \in C^{\infty} (R^{n})$ and $- Δ N h = h$ : after writing $N h (x) = \int Φ (z) h (x - z) d z$ , differentiation falls on $h$ , whose translated supports for $x$ in a compact set lie in one bounded set. Local integrability of $Φ$ dominates every such differentiated integrand, so $D^{β} N h = N (D^{β} h)$ for all $β$ ; and if $v$ is locally integrable and weakly harmonic on an open set, then $v$ is represented by a smooth function there. (Newtonian potential of compactly supported data, Newtonian potentials solve the distributional Poisson equation, Hölder data give a classical Newtonian solution, Locally integrable weakly harmonic functions are smooth)

[F7]

Maximum principle and barrier (Weak maximum principle for the laplacian): if $v \in C^{2} (Ω) \cap C (\overset{ˉ}{Ω})$ and $Δ v \geq 0$ then $\max_{\overset{ˉ}{Ω}} v = \max_{\partial Ω} v$ . If $Ω \subseteq B_{R} (x_{0})$ and $q (x) : = M (R^{2} - ∣ x - x_{0} ∣^{2}) / (2 n)$ , then $Δ q = - M$ and $q \geq 0$ on $\overset{ˉ}{Ω}$ .

[F8]

Boundary Schauder estimate (Boundary Schauder estimate for the Dirichlet problem): if $v \in C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ satisfies $Δ v = h$ pointwise with $h \in C^{0, α} (\overset{ˉ}{Ω})$ and $v = 0$ on $\partial Ω$ , then $∥ v ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C (∥ v ∥_{C^{0} (Ω)} + ∥ h ∥_{C^{0, α} (\overset{ˉ}{Ω})})$ with $C = C (n, α, Ω)$ (the coefficients are constant, so $λ = Λ = 1$ , $K = M = 0$ ).

[F9]

A pointwise bounded equicontinuous sequence of continuous maps from the compact metric space $\overset{ˉ}{Ω}$ to a finite-dimensional Euclidean space has a uniformly convergent subsequence (Real and finite-dimensional Euclidean Ascoli–Arzelà criteria). The identification of uniform limits of derivatives is proved locally in step 6.1, using the fundamental theorem of calculus on balls compactly contained in $Ω$ ; the uniform Holder bound for the Hessians passes to the limit pointwise.

Proof

technique · direct

1.1F1givenalgebraA1

Reduction to zero boundary values. Put $F : = f + Δ g$ on $\overset{ˉ}{Ω}$ . Since $g \in C^{2, α} (\overset{ˉ}{Ω})$ and $Δ g \in C^{0, α} (\overset{ˉ}{Ω})$ , one has $F \in C^{0, α} (\overset{ˉ}{Ω})$ with $∥ F ∥_{C^{0, α}} \leq ∥ f ∥_{C^{0, α}} + C (Ω) ∥ g ∥_{C^{2, α}}$ . If $u$ is the weak solution of the problem of [F1], then $w : = u - g$ lies in $H_{0}^{1} (Ω)$ and, for every $φ \in H_{0}^{1} (Ω)$ , $\int \nabla w \cdot \nabla φ = \int \nabla u \cdot \nabla φ - \int \nabla g \cdot \nabla φ = \int f φ + \int Δ g φ = \int F φ$ , the middle identity for $g$ being the weak form of $- Δ g$ for $C^{2}$ functions (approximate $φ$ by $C_{c}^{\infty}$ functions and integrate by parts). So $w$ is a weak solution of the zero-boundary problem $- Δ w = F$ ; conversely, if such a $w$ is shown to be $C^{2, α}$ up to the boundary with $w = 0$ on $\partial Ω$ , then $u = w + g$ is the required solution. It suffices to prove the zero-boundary statement for $F \in C^{0, α} (\overset{ˉ}{Ω})$ : find $w \in H_{0}^{1} (Ω)$ with $- Δ w = F$ weakly, $w \in C^{2, α} (\overset{ˉ}{Ω})$ and $∥ w ∥_{C^{2, α}} \leq C ∥ F ∥_{C^{0, α}}$ .

2.1F2F3step 1.1algebra

Extending and mollifying the data. By [F2] extend $F$ to $F^{♯} \in C_{c}^{0, α} (R^{n})$ with $∥ F^{♯} ∥_{C^{0, α}} \leq C_{1} ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})}$ , and put $F_{ε} : = F^{♯} * ρ_{ε}$ as in [F3]; then $M : = \sup_{ε} ∥ F_{ε} ∥_{\infty} \leq ∥ F^{♯} ∥_{\infty} \leq C_{1} ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})}$ and $\sup_{ε} [F_{ε}]_{0, α} \leq C_{1} ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})}$ , while $F_{ε} \to F$ uniformly on $\overset{ˉ}{Ω}$ .

3.1F1step 2.1algebra

The approximating weak solutions and the initial energy bound. Choose $p > n$ , for instance $p = n + 1$ . For each $ε$ , $F_{ε} ∣_{Ω} \in L^{p} (Ω) \subset H^{- 1} (Ω)$ , so [F1] gives a unique weak solution $w_{ε} \in H_{0}^{1} (Ω)$ of $- Δ w_{ε} = F_{ε}$ . Testing the weak equation with $w_{ε}$ (or its complex conjugate) and using Poincar'e gives $∥ w_{ε} ∥_{H^{1}} \leq C ∥ F_{ε} ∥_{L^{2}} \leq C ∥ F ∥_{C^{0, α}}$ , uniformly in $ε$ .

4.1F1F4F5step 2.1step 3.1algebra

Uniform $W^{2, p}$ bound, including the $L^{p}$ term. Use the finite-exponent bootstrap in the proof of [F4], not just its final a priori estimate. For $n > 2$ set $q_{0} = \min {p, 2 n / (n - 2)}$ ; for $n = 2$ set $q_{0} = p$ . The energy estimate of step 3.1 and the first-order Sobolev embedding control $∥ w_{ε} ∥_{L^{q_{0}}}$ . The supplier proof chooses a fixed shift $λ$ and a finite list $q_{0} \leq q_{1} < \dots < q_{m} = p$ (with no further step when $q_{0} = p$ ), where $q_{i + 1} = \min {p, n q_{i} / (n - 2 q_{i})}$ while $2 q_{i} < n$ , and $q_{i + 1} = p$ once $2 q_{i} \geq n$ . At the first exponent, the shifted strong-solvability estimate for $(λ - Δ) z = F_{ε} + λ w_{ε}$ , together with energy uniqueness, identifies $z = w_{ε}$ and bounds $∥ w_{ε} ∥_{W^{2, q_{0}}}$ by $C (∥ F_{ε} ∥_{L^{q_{0}}} + ∥ w_{ε} ∥_{L^{q_{0}}})$ . At each later exponent $q_{i + 1}$ , the embedding in [F5] bounds $∥ w_{ε} ∥_{L^{q_{i + 1}}}$ by the preceding $W^{2, q_{i}}$ norm; the next shifted estimate and energy uniqueness then give the $W^{2, q_{i + 1}}$ bound. Every $∥ F_{ε} ∥_{L^{q_{i}}}$ is bounded by $C ∥ F ∥_{C^{0, α}}$ , and the list is finite, so induction gives $∥ w_{ε} ∥_{W^{2, p}} \leq C (∥ F_{ε} ∥_{L^{p}} + ∥ w_{ε} ∥_{H^{1}}) \leq C ∥ F ∥_{C^{0, α}}$ , uniformly in $ε$ . This controls the $L^{p} (w_{ε})$ term left explicit in the supplier's final a priori estimate. Applying [F5] with $k = 2$ , $p > n$ , each $w_{ε}$ has a $C^{1, γ} (\overset{ˉ}{Ω})$ representative, for any fixed $0 < γ < 1 - n / p$ , with uniformly bounded norm. Its trace is zero because $w_{ε} \in H_{0}^{1} (Ω)$ ; [F1] identifies this trace with the boundary values of the continuous representative.

4.2F6step 3.1algebra

Interior smoothness. Fix a point $x_{0} \in Ω$ and a ball $B ⋐ Ω$ around it. The function $F_{ε}$ is smooth near $\overset{ˉ}{B}$ ; choose $θ \in C_{c}^{\infty} (Ω)$ with $θ = 1$ on a neighbourhood of $\overset{ˉ}{B}$ and set $H_{ε} : = θ F_{ε}$ , a compactly supported smooth function. By [F6], $N H_{ε} \in C^{\infty} (R^{n})$ and $- Δ N H_{ε} = H_{ε} = F_{ε}$ on $B$ . Hence $Δ (w_{ε} - N H_{ε}) = 0$ weakly on $B$ : for every $φ \in C_{c}^{\infty} (B)$ , $\int \nabla (w_{ε} - N H_{ε}) \cdot \nabla φ = \int F_{ε} φ - \int F_{ε} φ = 0$ . By the local smoothness of weakly harmonic functions in [F6], $w_{ε} - N H_{ε}$ agrees on $B$ with a smooth function; since $N H_{ε}$ is smooth, $w_{ε}$ agrees on $B$ with a smooth function. As $B$ and $x_{0}$ were arbitrary, $w_{ε} \in C^{\infty} (Ω)$ , and it satisfies $- Δ w_{ε} = F_{ε}$ pointwise in $Ω$ .

5.1F7step 2.1step 4.1step 4.2algebra

A uniform supremum bound with the weak maximum principle's sign. Choose $R > 0$ and $x_{0}$ with $Ω \subseteq B_{R} (x_{0})$ and put $q (x) : = M (R^{2} - ∣ x - x_{0} ∣^{2}) / (2 n)$ , where $M : = \sup_{ε} ∥ F_{ε} ∥_{\infty}$ . Then $q \geq 0$ on $\overset{ˉ}{Ω}$ and $Δ q = - M$ . For a real-valued solution component $v_{ε}$ with datum $F_{ε}$ satisfying $∣ F_{ε} ∣ \leq M$ , one has $Δ (v_{ε} - q) = M - F_{ε} \geq 0, Δ (- v_{ε} - q) = M + F_{ε} \geq 0.$ Both comparison functions are in $C^{2} (Ω) \cap C (\overset{ˉ}{Ω})$ by steps 4.1 and 4.2, and their boundary values are $- q \leq 0$ . The weak maximum principle [F7] therefore gives $v_{ε} - q \leq 0$ and $- v_{ε} - q \leq 0$ , hence $∣ v_{ε} ∣ \leq q \leq M R^{2} / (2 n)$ . If the data are complex, apply this argument to the real and imaginary parts separately; then $∣ w_{ε} ∣ \leq \sqrt{2} M R^{2} / (2 n)$ . By step 2.1, $M \leq C ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})}$ , so this is a uniform $C^{0}$ bound.

6.1F8F9step 5.1algebra

Uniform $C^{2, α}$ bounds and the limit. The functions $w_{ε}$ lie in $C^{2} (Ω) \cap C^{0} (\overset{ˉ}{Ω})$ by steps 4.1 and 4.2, vanish on $\partial Ω$ , and satisfy $- Δ w_{ε} = F_{ε}$ pointwise with $F_{ε} \in C^{0, α} (\overset{ˉ}{Ω})$ and $∥ F_{ε} ∥_{C^{0, α}} \leq C_{1} ∥ F ∥_{C^{0, α}}$ . Apply [F8] to each real component of $w_{ε}$ with right-hand side the corresponding component of $- F_{ε}$ (the estimate is unchanged by this sign), and combine the component bounds if the data are complex. Using step 5.1, $∥ w_{ε} ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C_{7} (∥ w_{ε} ∥_{C^{0} (Ω)} + ∥ F_{ε} ∥_{C^{0, α} (\overset{ˉ}{Ω})}) \leq C_{8} ∥ F ∥_{C^{0, α} (\overset{ˉ}{Ω})},$ uniformly in $ε$ . The boundary Schauder estimate gives uniform $C^{2, α}$ control in each member of a finite cover of $\overset{ˉ}{Ω}$ by interior balls and flattened boundary half-boxes. Thus the function, gradient and Hessian components are equicontinuous and pointwise bounded on $\overset{ˉ}{Ω}$ ; if the functions are complex, list their real and imaginary components separately. By [F9], a subsequence of this finite-dimensional vector-valued family converges uniformly to limits $(w, G, H)$ . On every ball $B ⋐ Ω$ , the fundamental theorem of calculus along segments in $B$ and uniform convergence give $D w = G$ and $D G = H$ . The limits $G, H$ are continuous on $\overset{ˉ}{Ω}$ , so these derivatives extend continuously to the boundary; pointwise convergence of the Hessian difference quotients gives $[D^{2} w]_{0, α; Ω} \leq {lim inf}_{ε} [D^{2} w_{ε}]_{0, α; Ω}$ . Hence $w \in C^{2, α} (\overset{ˉ}{Ω})$ with the stated bound. Uniform convergence of $F_{ε}$ and of the second derivatives gives $- Δ w = F$ pointwise in $Ω$ and $w = 0$ on $\partial Ω$ .

7.1step 1.1step 6.1F1algebra

Identification with the weak solution. The limit $w \in C^{2} (\overset{ˉ}{Ω})$ with $w = 0$ on $\partial Ω$ satisfies $\int \nabla w \cdot \nabla φ = \int F φ$ for every $φ \in H_{0}^{1} (Ω)$ : for $φ \in C_{c}^{\infty} (Ω)$ this is integration by parts, and $C_{c}^{\infty} (Ω)$ is dense in $H_{0}^{1} (Ω)$ . By [F1] the weak solution of the zero-boundary problem is unique, so $w$ is the unique weak solution $w = u - g$ of the original problem as identified in step 1.1. Therefore $u = w + g \in C^{2, α} (\overset{ˉ}{Ω})$ solves $- Δ u = f$ pointwise and $u = g$ on $\partial Ω$ , and $∥ u ∥_{C^{2, α}} \leq ∥ w ∥_{C^{2, α}} + ∥ g ∥_{C^{2, α}} \leq C ∥ F ∥_{C^{0, α}} + C ∥ g ∥_{C^{2, α}} \leq C^{'} (∥ f ∥_{C^{0, α}} + ∥ g ∥_{C^{2, α}})$ by step 1.1. This is the displayed estimate of the statement.

8.1step 2.1step 6.1step 7.1F1F8given∎

Conclusion. The weak Dirichlet problem has exactly one solution by [F1], and steps 2.1-7.1 show that this solution is the limit of the smooth approximating solutions, is of class $C^{2, α} (\overset{ˉ}{Ω})$ with the stated bound, and solves the equation classically. In particular the Laplace operator with Dirichlet boundary values on a bounded $C^{2, α}$ domain has the Schauder a priori estimate on weak solutions, a fact used as the base point of the method of continuity. The proof uses only the fixed exponent $p > n$ in the weak-to-strong step, and the finiteness of all constants is uniform in the mollification parameter.

Remarks

The proof is the classical approximation scheme: solve smooth approximating problems weakly, upgrade them with $W^{2, p}$ regularity, embed, use the maximum principle for a uniform supremum bound, apply the boundary Schauder estimate and pass to the limit by Arzela-Ascoli. Uniqueness of the weak solution identifies the limit, so no subsequence ambiguity remains.
The uniform sup bound is what makes the boundary Schauder estimate applicable with constants independent of $ε$ ; the result is an a posteriori (regularity) statement, while the a priori estimate in the boundary theorem is applied after its quoted input establishes closure regularity.
Only the fixed pair $(p, γ)$ with $p = n + 1$ , $0 < γ < 1 - n / p$ is used in the embedding step; any $p > n$ gives the same conclusion.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Global Schauder estimate and classical Dirichlet solvability by the continuity method

Statement

Assume the Axiom of Choice and Countable Choice. Let $n \geq 2$ , $0 < α < 1$ , let $Ω$ be a bounded $C^{2, α}$ domain and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic on $\overset{ˉ}{Ω}$ , with $a^{i j}, b^{i}, c \in C^{0, α} (\overset{ˉ}{Ω})$ , constants $λ, Λ$ , $[A]_{0, α} \leq K$ , and $∥ b ∥_{C^{0, α}} + ∥ c ∥_{C^{0, α}} \leq M$ . Put $X : = {u \in C^{2, α} (\overset{ˉ}{Ω}) : u ∣_{\partial Ω} = 0}$ and $L_{t} : = t L + (1 - t) Δ$ for $t \in [0, 1]$ . Assume that each $L_{t} : X \to C^{0, α} (\overset{ˉ}{Ω})$ is injective. Then every $L_{t}$ is bijective; in particular every $f \in C^{0, α} (\overset{ˉ}{Ω})$ and $g \in C^{2, α} (\overset{ˉ}{Ω})$ determine a unique classical solution of $L u = f$ in $Ω$ , $u = g$ on $\partial Ω$ , and $∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C (∥ f ∥_{C^{0, α} (\overset{ˉ}{Ω})} + ∥ g ∥_{C^{2, α} (\overset{ˉ}{Ω})}),$ where $C$ is uniform in $t, u, f, g$ .

Facts & Assumptions

Given: the Axiom of Choice and Countable Choice, $n \geq 2$ , $0 < α < 1$ , the bounded $C^{2, α}$ domain $Ω$ , the operator $L$ with the stated coefficient bounds, the family $L_{t} = t L + (1 - t) Δ$ , and the hypothesis that every $L_{t}$ is injective on $X$ .

[A1]

The Axiom of Choice and Countable Choice are used through the Banach-space, maximum-principle, Arzela-Ascoli and Schauder-regularity inputs; the further choices in the contradiction argument are finite or sequential. (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The closure class $C^{2, α} (\overset{ˉ}{Ω})$ and its zero-boundary subspace $X$ are Banach spaces, as is $Y : = C^{0, α} (\overset{ˉ}{Ω})$ , with the full finite Hölder norms. These are precisely the boundary-extension classes and closed subspaces of The closure Hölder spaces are Banach spaces; no identification with all of $C_{b}^{2, α} (Ω)$ is needed.

[F2]

Each $L_{t}$ maps $X$ boundedly into $Y$ , with a bound uniform in $t$ : $∥ L_{t} u ∥_{C^{0, α}} \leq C_{L} ∥ u ∥_{C^{2, α}}$ for $u \in X$ and $t \in [0, 1]$ , because the coefficients are bounded in $C^{0, α}$ and the principal matrices $A_{t} = t A + (1 - t) I$ are uniformly elliptic with constants $\min {λ, 1}, \max {Λ, 1}$ , $C^{0, α}$ seminorm at most $K$ and lower-order coefficient bounds at most $M$ . Moreover $t \mapsto L_{t}$ is affine, so $L_{t} - L_{s} = (t - s) (L - Δ)$ with $∥ (L - Δ) u ∥_{C^{0, α}} \leq C_{L}^{'} ∥ u ∥_{C^{2, α}}$ . (Uniformly elliptic nondivergence-form operators and their frozen coefficients)

[F3]

Uniform boundary Schauder estimate (Boundary Schauder estimate for the Dirichlet problem): applied to $L_{t}$ with the uniform constants of [F2], it gives $∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C_{1} (∥ L_{t} u ∥_{C^{0, α} (\overset{ˉ}{Ω})} + ∥ u ∥_{C^{0} (Ω)}) (u \in X, t \in [0, 1]),$ with $C_{1}$ depending only on $n, α$ , the uniform ellipticity and coefficient bounds and $Ω$ .

[F4]

Compactness: a sequence bounded in $X$ has a subsequence converging in $C^{2} (\overset{ˉ}{Ω})$ ; this is the vector-valued Arzela-Ascoli theorem Real and finite-dimensional Euclidean Ascoli–Arzelà criteria applied to the maps $x \mapsto (u_{j} (x), \nabla u_{j} (x), D^{2} u_{j} (x))$ , which are equicontinuous and pointwise bounded because $∥ u_{j} ∥_{C^{2, α}} \leq 1$ . (Arzelà--Ascoli for real $C (K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded)

[F5]

Base point: $L_{0} = Δ$ is bijective from $X$ to $Y$ . Injectivity: if $Δ u = 0$ on $Ω$ with $u = 0$ on $\partial Ω$ , the weak maximum principle (applied to $u$ and $- u$ , componentwise for complex functions) gives $u = 0$ . Surjectivity: given $h \in Y$ , put $f : = - h \in C^{0, α} (\overset{ˉ}{Ω})$ and let $u$ be the weak solution of $- Δ u = f$ with zero boundary values given by Global Schauder regularity for the weak Dirichlet Laplacian; then $u \in C^{2, α} (\overset{ˉ}{Ω})$ , $u = 0$ on $\partial Ω$ and $Δ u = h$ pointwise, so $L_{0} u = h$ . (Weak maximum principle for the laplacian)

[F6]

Method of continuity (The method of continuity for a uniformly estimated affine family of bounded operators): if $L_{0}$ is bijective and, for some $0 \leq C < \infty$ , $∥ x ∥_{X} \leq C ∥ L_{t} x ∥_{Y}$ holds for all $t \in [0, 1]$ and all $x \in X$ , then every $L_{t}$ is bijective with $∥ L_{t}^{- 1} ∥ \leq C$ .

Proof

technique · direct

1.1F1F2givenA1

Setting. By [F1], $X$ and $Y$ are Banach spaces over the same field, and by [F2] each $L_{t}$ is a bounded operator $X \to Y$ forming an affine family $L_{t} = (1 - t) L_{0} + t L_{1}$ with $L_{0} = Δ$ and $L_{1} = L$ . It remains to verify the two hypotheses of [F6]: the bijectivity of $L_{0}$ and the uniform a priori estimate.

1.2F2F3

The estimate with the supremum term. By [F3], for every $u \in X$ and $t \in [0, 1]$ , $∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})} \leq C_{1} (∥ L_{t} u ∥_{C^{0, α} (\overset{ˉ}{Ω})} + ∥ u ∥_{C^{0} (Ω)}) .$ This is the only place where the boundary Schauder estimate enters; its constant is uniform in $t$ because the family is uniformly elliptic with uniformly bounded $C^{0, α}$ coefficients.

2.1step 1.2F2F4givencontradiction

Removing the supremum term. Suppose the uniform estimate $∥ u ∥_{X} \leq C ∥ L_{t} u ∥_{Y}$ failed for every finite $C$ . Then for each $j \in N$ there are $t_{j} \in [0, 1]$ and $u_{j} \in X$ with $∥ u_{j} ∥_{C^{2, α}} = 1$ and $∥ L_{t_{j}} u_{j} ∥_{C^{0, α}} < 1 / (j + 1)$ . By step 1.2, $1 \leq C_{1} (1 / (j + 1) + ∥ u_{j} ∥_{C^{0}})$ , so $∥ u_{j} ∥_{C^{0}} \geq 1 / (2 C_{1})$ for all large $j$ . By [F4] and compactness of $[0, 1]$ there is a subsequence, relabelled, with $t_{j} \to t$ and $u_{j} \to u$ in $C^{2} (\overset{ˉ}{Ω})$ ; then $∥ u ∥_{C^{0}} = \lim ∥ u_{j} ∥_{C^{0}} \geq 1 / (2 C_{1}) > 0$ , so $u \neq 0$ , and $u ∣_{\partial Ω} = 0$ because $u_{j} ∣_{\partial Ω} = 0$ and the convergence is uniform. Moreover $L_{t} u = 0$ : indeed $L_{t_{j}} u_{j} = L_{t} u_{j} + (t_{j} - t) (L - Δ) u_{j}$ ; the second term tends to $0$ in the $C^{0, α}$ norm by [F2] and $∣ t_{j} - t ∣ \to 0$ with $∥ u_{j} ∥_{C^{2, α}} = 1$ , while $L_{t} u_{j} \to L_{t} u$ in the supremum norm because $u_{j} \to u$ in $C^{2}$ and the coefficients of $L_{t}$ are fixed continuous functions; since $L_{t_{j}} u_{j} \to 0$ in $Y$ , it follows that $L_{t} u = 0$ . For distinct $x, y$ , pass the uniformly bounded Hessian difference quotients to the $C^{2}$ limit to obtain $[D^{2} u]_{0, α} \leq {lim inf}_{j} [D^{2} u_{j}]_{0, α} < \infty$ . Hence $u \in X$ , so the injectivity hypothesis on $L_{t}$ forces $u = 0$ , contradicting $u \neq 0$ . Hence there is $0 < C < \infty$ with $∥ u ∥_{C^{2, α}} \leq C ∥ L_{t} u ∥_{C^{0, α}}$ for all $u \in X$ and $t \in [0, 1]$ .

3.1F5step 2.1

The base point is bijective. By [F5], $L_{0} = Δ$ is injective and surjective, hence bijective, with $∥ L_{0}^{- 1} ∥ \leq C$ already implied by the uniform estimate of step 2.1.

4.1step 2.1step 3.1F6

The method of continuity. Applying [F6] with $L_{0} = Δ$ , $L_{1} = L$ , the uniform estimate of step 2.1 and the bijectivity of step 3.1, every $L_{t} : X \to Y$ is bijective and $∥ L_{t}^{- 1} ∥_{Y \to X} \leq C$ with the same constant $C$ for all $t \in [0, 1]$ .

5.1step 4.1F2algebra

Nonzero boundary data. Let $f \in Y$ , $g \in C^{2, α} (\overset{ˉ}{Ω})$ and fix $t \in [0, 1]$ . Since $L_{t} g \in Y$ and $L_{t}$ is bijective by step 4.1, there is a unique $u_{0} \in X$ with $L_{t} u_{0} = f - L_{t} g$ ; then $u : = u_{0} + g$ lies in $C^{2, α} (\overset{ˉ}{Ω})$ , satisfies $L_{t} u = f$ in $Ω$ and $u = g$ on $\partial Ω$ , and $∥ u ∥_{C^{2, α}} \leq ∥ u_{0} ∥_{C^{2, α}} + ∥ g ∥_{C^{2, α}} \leq C ∥ f - L_{t} g ∥_{C^{0, α}} + ∥ g ∥_{C^{2, α}} \leq C^{'} (∥ f ∥_{C^{0, α}} + ∥ g ∥_{C^{2, α}})$ by [F2], with $C^{'}$ independent of $t$ and of $(u, f, g)$ . Uniqueness for fixed $t$ follows from injectivity: two solutions differ by an element of $X$ in the kernel of $L_{t}$ .

6.1step 4.1step 5.1F6given∎

Conclusion. Under the stated injectivity hypothesis, the affine family $L_{t}$ satisfies the uniform a priori estimate of step 2.1 and has the bijective base point $L_{0} = Δ$ of step 3.1; the method of continuity therefore makes every $L_{t}$ bijective, uniformly in $t$ , and subtracting a $C^{2, α}$ extension of the boundary datum produces the classical solution of the Dirichlet problem for $L$ with the displayed estimate. In particular the injectivity hypothesis can be verified separately for each $t$ (a separate uniqueness argument must respect the displayed positive-principal-part sign convention), and the conclusion is a genuine existence statement for classical solutions, obtained without compactness of the operator $L$ itself.

Remarks

The two structural inputs are the boundary Schauder estimate, which supplies the uniform a priori bound, and the weak solvability of the Dirichlet Laplacian (through the maximum principle and the global Schauder regularity theorem), which supplies the bijective base point. The contradiction step uses Arzela-Ascoli to rule out a loss of the supremum term.
The constant is uniform in $t$ because the uniform coefficient bounds and injectivity on the fixed compact parameter family give the estimate in step 2.1; the theorem does not use symmetry of $L$ , and the injectivity hypothesis is the exact place where a possible eigenvalue of the family is excluded.

CorollaryStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Injectivity removes the $L^{p}$ term from the global $W^{2, p}$ estimate

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , $1 < p < \infty$ , let $Ω$ be a bounded $C^{1, 1}$ domain and let $L = a^{i j} \partial_{i} \partial_{j} + b^{i} \partial_{i} + c$ be uniformly elliptic with $a^{i j} \in C^{0} (\overset{ˉ}{Ω})$ , $b, c \in L^{\infty} (Ω)$ as in Global $W^{2, p}$ Dirichlet estimate on a $C^{1, 1}$ domain. Assume that the homogeneous Dirichlet problem has only the trivial strong solution: if $w \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ and $L w = 0$ almost everywhere, then $w = 0$ . Then there is $C < \infty$ with $∥ u ∥_{W^{2, p} (Ω)} \leq C ∥ L u ∥_{L^{p} (Ω)} for every u \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω) .$ No symmetry of $L$ is used and no spectral hypothesis beyond the stated injectivity enters; the constant can additionally depend on the particular operator $L$ through its separation from a nontrivial Dirichlet kernel. Injectivity alone supplies no bound uniform over all operators with the same coefficient upper bounds. The Axiom of Choice is needed because the compactness alternatives of Rellich--Kondrachov are invoked.

Facts & Assumptions

Given: the Axiom of Choice, $n \geq 2$ , $1 < p < \infty$ , the bounded $C^{1, 1}$ domain $Ω$ , the operator $L$ with the stated coefficient bounds, the injectivity hypothesis, and the a priori estimate of Global $W^{2, p}$ Dirichlet estimate on a $C^{1, 1}$ domain.

[A1]

The Axiom of Choice is the standing hypothesis; it is inherited by the a priori estimate and Sobolev completeness, and used through the compactness and extension theorems that make $Ω$ a bounded extension domain. (The Axiom of Choice)

[F1]

A priori estimate (Global $W^{2, p}$ Dirichlet estimate on a $C^{1, 1}$ domain): there is $C_{0}$ with $∥ v ∥_{W^{2, p} (Ω)} \leq C_{0} (∥ L v ∥_{L^{p} (Ω)} + ∥ v ∥_{L^{p} (Ω)})$ for all $v \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ .

[F2]

Compactness alternatives for a bounded extension domain $Ω$ : for $1 \leq p < n$ every bounded sequence in $W^{1, p} (Ω)$ has a subsequence converging in $L^{p} (Ω)$ (The Rellich--Kondrachov theorem for $1 \leq p < n$ on bounded extension domains with $q = p < p^{*}$ ); for $p = n$ every bounded sequence in $W^{1, n} (Ω)$ has a subsequence converging in $L^{q} (Ω)$ for each fixed finite $q$ , in particular $q = n$ (Rellich--Kondrachov at the critical source exponent $p = n$ ); for $p > n$ every bounded sequence in $W^{1, p} (Ω)$ has a subsequence converging in $L^{q} (Ω)$ for every $1 \leq q < \infty$ , in particular $q = p$ (Morrey--Rellich compactness for $p > n$ ).

[F3]

A bounded $C^{1, 1}$ domain is a bounded Lipschitz extension domain for $W^{1, p}$ : there is a bounded extension operator $W^{1, p} (Ω) \to W^{1, p} (R^{n})$ (Bounded C^k domains and boundary charts, Bounded C^k domains admit integer-order Sobolev extension). This is the only extension input used here, to verify that the Rellich--Kondrachov results in [F2] apply; no $W^{k, q}$ extension for $k \geq 2$ is needed.

[F4]

On the subspace $W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ , which is closed in $W^{2, p} (Ω)$ the operator $L$ is bounded into $L^{p} (Ω)$ : $∥ L v ∥_{L^{p}} \leq C (Λ, M) ∥ v ∥_{W^{2, p}}$ for $v \in W^{2, p} (Ω)$ ; the space $W_{0}^{1, p} (Ω)$ is closed in $W^{1, p} (Ω)$ and hence in $W^{2, p} (Ω)$ . (Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure)

Proof

technique · direct

1.1F3F4givenA1

Contradiction setup and a bounded sequence. Suppose the inequality fails: then for every integer $j \geq 1$ there is $u_{j} \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ with $∥ u_{j} ∥_{W^{2, p}} = 1$ and $∥ L u_{j} ∥_{L^{p}} < 1 / j$ ; equivalently, after rescaling, a sequence with $∥ u_{j} ∥_{W^{2, p}} = 1$ and $∥ L u_{j} ∥_{L^{p}} \to 0$ . For each $j$ the $W^{1, p}$ norm is bounded by the $W^{2, p}$ norm up to constants, so $(u_{j})$ is bounded in $W^{1, p} (Ω)$ , and by [F3] the domain $Ω$ is a bounded extension domain for $W^{1, p}$ .

2.1F2step 1.1

An $L^{p}$ -convergent subsequence. By [F2] applied to the bounded sequence $(u_{j})$ , in each of the three cases $p < n$ , $p = n$ , $p > n$ there is a subsequence, relabelled $(u_{j})$ , converging in $L^{p} (Ω)$ to some $w \in L^{p} (Ω)$ . In the case $p < n$ the admissible exponents form the interval $[1, p^{*})$ and $q = p$ is admissible; in the case $p = n$ every finite $q$ is admissible and $q = n = p$ ; in the case $p > n$ every $1 \leq q < \infty$ is admissible and $q = p$ .

3.1step 1.1step 2.1F1algebra

Cauchy in $W^{2, p}$ via the a priori estimate. Apply [F1] to the differences $u_{j} - u_{k} \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ : $∥ u_{j} - u_{k} ∥_{W^{2, p}} \leq C_{0} (∥ L (u_{j} - u_{k}) ∥_{L^{p}} + ∥ u_{j} - u_{k} ∥_{L^{p}}) \leq C_{0} (∥ L u_{j} ∥_{L^{p}} + ∥ L u_{k} ∥_{L^{p}} + ∥ u_{j} - u_{k} ∥_{L^{p}}) .$ The first two terms tend to $0$ by construction, and the third by the $L^{p}$ convergence of step 2.1; hence $(u_{j})$ is Cauchy in $W^{2, p} (Ω)$ , which is complete by Integer-order Sobolev spaces are Banach, and converges to some $v \in W^{2, p} (Ω)$ with $v$ equal to the $L^{p}$ -limit $w$ of step 2.1.

4.1step 3.1F4given

The limit is a vanishing strong solution. The space $W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ is closed in $W^{2, p} (Ω)$ by [F4], so $v$ belongs to it; the boundedness of $L : W^{2, p} (Ω) \to L^{p} (Ω)$ and $L u_{j} \to 0$ in $L^{p}$ give $L v = 0$ almost everywhere. By the injectivity hypothesis $v = 0$ .

5.1step 1.1step 2.1step 3.1step 4.1F1given

Contradiction. Applying [F1] to $u_{j}$ and using the normalization, $1 = ∥ u_{j} ∥_{W^{2, p}} \leq C_{0} (∥ L u_{j} ∥_{L^{p}} + ∥ u_{j} ∥_{L^{p}}) ⟶ 0$ because $∥ L u_{j} ∥_{L^{p}} \to 0$ and $∥ u_{j} ∥_{L^{p}} \to ∥ v ∥_{L^{p}} = 0$ by steps 2.1 and 4.1. This contradiction shows that the failure assumed in step 1.1 is impossible, that is, there is $C$ with $∥ u ∥_{W^{2, p} (Ω)} \leq C ∥ L u ∥_{L^{p} (Ω)}$ for all $u \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ .

6.1step 5.1F1F2F3given∎

Conclusion. Assume that the only strong solution of $L w = 0$ in $W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ is $w = 0$ . Then the compactness of the Sobolev embedding upgrades the a priori estimate [F1] to the pure $L^{p}$ estimate displayed in the statement, the constant absorbing the $L^{p}$ term through the contradiction argument. No symmetry, self-adjointness or spectral hypothesis on $L$ is used, and the only choice principle invoked is the Axiom of Choice, including its inherited uses in the a priori estimate, completeness, Rellich--Kondrachov and extension theorems.

Remarks

The structure is the classical one: a priori estimate plus compactness turns injectivity of the homogeneous problem into the sharper estimate without the $L^{p}$ term. The compactness is used only to extract an $L^{p}$ -convergent subsequence; the $W^{2, p}$ convergence is then produced by the estimate itself.
The three Rellich--Kondrachov branches are the reason the corollary assumes the Axiom of Choice, and the a priori estimate used here also assumes Choice through its trace and extension suppliers. If one of the suppliers were only available under a weaker principle, the corresponding branch would have to be stated separately.

CorollaryStatement: Literature-sourcedProof: AI-adaptedOpen item page →

$W^{2, p}$ regularity implies classical or H"older regularity when $p$ is large

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , $1 n / 2$ , every $u \in W^{2, p} (Ω)$ has a representative in $C^{0, γ} (\overset{ˉ}{Ω})$ for every $0 < γ < \min {1, 2 - n / p}$ , with the norm controlled by $∥ u ∥_{W^{2, p}}$ . (ii) If $p > n$ , every such $u$ has a representative in $C^{1, γ} (\overset{ˉ}{Ω})$ for every $0 < γ < 1 - n / p$ , with the corresponding norm bound; in particular $D^{2} u \in L^{p}$ . For any nondivergence expression $L u = \sum_{∣ β ∣ \leq 2} a_{β} D^{β} u$ with $a_{β} \in L^{\infty} (Ω)$ , this gives $L u \in L^{p} (Ω)$ ; whenever $L u = f$ also holds distributionally for some $f \in L^{p} (Ω)$ , the equality then holds almost everywhere. (iii) If $p > n / (1 - α)$ for $0 < α < 1$ , the representative is $C^{1, α} (\overset{ˉ}{Ω})$ . No finite $p$ gives $C^{2, α}$ regularity in general: on a ball, the function $u (x) = (x_{1})_{+}^{2}$ belongs to $W^{2, p}$ for every finite $p$ , while $D_{11} u = 21_{{x_{1} > 0}}$ has no continuous representative.

Facts & Assumptions

Given: the Axiom of Choice, $n \geq 2$ , $1 < p < \infty$ , the bounded $W^{2, p}$ -extension domain $Ω$ , and $u \in W^{2, p} (Ω)$ .

[A1]

The Axiom of Choice is the standing hypothesis, used through the extension operator and the embedding theorem below. (The Axiom of Choice)

[F1]

By the extension property there is a bounded linear $E : W^{2, p} (Ω) \to W^{2, p} (R^{n})$ with $(E u) ∣_{Ω} = u$ almost everywhere and $∥ E u ∥_{W^{2, p} (R^{n})} \leq C_{E} ∥ u ∥_{W^{2, p} (Ω)}$ . (Sobolev extension domains and extension operators)

[F2]

Higher-order Sobolev embedding (Higher-order Sobolev embedding): for a bounded extension domain and $k \geq 1$ , $1 \leq q < \infty$ , (a) if $k q < n$ then $W^{k, q} ↪ L^{r}$ for every $1 \leq r \leq n q / (n - k q)$ ; (b) if $k q = n$ then $W^{k, q} ↪ L^{r}$ for every finite $r$ ; (c) if $k q > n$ then every $u \in W^{k, q}$ has a representative in $C^{m, β} (\overset{ˉ}{Ω})$ for every integer $m \geq 0$ and $0 < β < 1$ with $m + β < k - n / q$ , with the norm bounded by a constant times $∥ u ∥_{W^{k, q}}$ . Applied with $k = 2$ and $q = p$ , the three cases are $2 p < n$ , $2 p = n$ , $2 p > n$ ; the case $2 p > n$ is exactly $p > n / 2$ . (Higher-order Sobolev embedding)

[F3]

The witness on the unit ball $Ω = B (0, 1)$ : for $u (x) = (x_{1})_{+}^{2} : = \max {x_{1}, 0}^{2}$ , the weak derivatives are $\partial_{1} u = 2 (x_{1})_{+}$ , $\partial_{11} u = 21_{{x_{1} > 0}}$ and $\partial_{i j} u = 0$ otherwise. Thus $u \in W^{2, p} (Ω)$ for every finite $p$ . On the disk section $Ω \cap {x_{1} = 0}$ the one-sided values of $\partial_{11} u$ differ, so this weak derivative has no continuous representative; consequently $u \notin C^{2, α} (\overset{ˉ}{Ω})$ for every $0 < α < 1$ . (Integer-order Sobolev spaces and their norms)

[F4]

The unit ball is a bounded $C^{2}$ domain and hence a $W^{2, p}$ -extension domain for every $1 \leq p \leq \infty$ (Bounded C^k domains and boundary charts, Bounded C^k domains admit integer-order Sobolev extension).

[F5]

If two locally integrable functions represent the same distribution on $Ω$ , they agree almost everywhere; this is the uniqueness of the zeroth weak derivative (Uniqueness of a weak derivative as an almost-everywhere class).

Proof

technique · direct application of the bounded-domain embedding

1.1F1F2givenA1

Domain hypothesis. By the definition of a $W^{2, p}$ -extension domain [F1], $Ω$ satisfies the bounded-domain premise of [F2] for $k = 2$ and exponent $p$ . The embedding conclusion of [F2] is already on $\overset{ˉ}{Ω}$ ; no embedding on the unbounded space $R^{n}$ is used.

1.2F3F4givenalgebra

The ball witness: no finite $p$ gives $C^{2, α}$ . On $Ω = B (0, 1)$ let $u (x) = \max {x_{1}, 0}^{2}$ . The function is $C^{1}$ with $\partial_{1} u = 2 \max {x_{1}, 0}$ , and integration by parts on the two sides of $x_{1} = 0$ gives the weak derivative $\partial_{11} u = 21_{{x_{1} > 0}}$ ; the interface term vanishes because $\max {x_{1}, 0}$ is continuous there. All second derivatives are bounded, so $u \in W^{2, p} (Ω)$ for every finite $p$ . If $\partial_{11} u$ had a continuous representative, it would equal $0$ on the negative open half-ball and $2$ on the positive open half-ball: the almost-everywhere equalities force these values on each open side by continuity. They cannot extend continuously across the interior disk $Ω \cap {x_{1} = 0}$ . Thus $u$ has no $C^{2, α}$ representative for any $0 < α < 1$ . The ball is in the stated extension-domain class by [F4].

2.1step 1.1F2algebra

Part (i): $p > n / 2$ . Then $2 p > n$ . For every $0 < γ < \min {1, 2 - n / p}$ the embedding [F2] with $k = 2$ , $q = p$ , $m = 0$ gives a representative $u^{*} \in C^{0, γ} (\overset{ˉ}{Ω})$ and $∥ u^{*} ∥_{C^{0, γ} (\overset{ˉ}{Ω})} \leq C ∥ u ∥_{W^{2, p} (Ω)}$ . If $2 - n / p \geq 1$ , the same strict inequality allows every $0 < γ < 1$ .

2.2step 1.1F2F5algebra

Part (ii): $p > n$ . Then $2 - n / p > 1$ . For each $0 < γ < 1 - n / p$ , one has $1 + γ < 2 - n / p$ , so [F2] with $k = 2$ , $q = p$ , $m = 1$ gives a representative $u^{*} \in C^{1, γ} (\overset{ˉ}{Ω})$ and the stated norm bound. The weak derivatives $D^{β} u$ with $∣ β ∣ \leq 2$ are $L^{p}$ classes by the definition of $W^{2, p}$ . Thus for $L = \sum_{∣ β ∣ \leq 2} a_{β} D^{β}$ with bounded coefficients, $L u$ is an $L^{p}$ class. If also $L u = f$ distributionally with $f \in L^{p}$ , then [F5] gives equality of the represented classes almost everywhere.

2.3step 1.1F2algebra

Part (iii): $p > n / (1 - α)$ . Then $1 + α < 2 - n / p$ , so [F2] with $k = 2$ , $q = p$ , $m = 1$ and exponent $α$ gives a representative in $C^{1, α} (\overset{ˉ}{Ω})$ with norm bounded by $C ∥ u ∥_{W^{2, p} (Ω)}$ . Parts (i), (ii), (iii) are direct applications of the bounded-domain higher-order embedding.

3.1step 2.1step 2.2step 2.3step 1.2F2∎

Conclusion. The higher-order embedding gives the asserted $C^{0, γ}$ representatives when $p > n / 2$ , $C^{1, γ}$ representatives when $p > n$ , and $C^{1, α}$ representatives when $p > n / (1 - α)$ . The ball witness of step 1.2 shows that no finite $p$ forces $C^{2, α}$ regularity in general, so the Sobolev and Schauder scales differ at the top order.

Remarks

The hypothesis is indexed by $(k, p) = (2, p)$ : a domain that is an extension domain for one pair need not be for another, and the statement uses only the displayed pair. The counterexample on the ball shows that the extension property alone, or any finite $p$ , cannot produce two H"older derivatives.
The strict exponent ranges in the statement are sufficient, rather than an assertion of optimality. For $p > n$ , the endpoint clause of Higher-order Sobolev embedding also gives $C^{1, 1 - n / p} (\overset{ˉ}{Ω})$ , since $2 - n / p \in (1, 2)$ is nonintegral; for example $p = 2 n$ gives $C^{1, 1 / 2}$ . The ball witness shows that no finite $p$ forces two Hölder derivatives.

RemarkRemark: Literature-sourcedProof: Not applicableOpen item page →

The Schauder and $W^{2, p}$ scales are different, not interchangeable

Remarks

On a bounded domain, $C^{2, α} (\overset{ˉ}{Ω})$ embeds strictly into $W^{2, p} (Ω)$ for every finite $p$ , and there is no reverse inclusion. Thus the spaces are not equivalent at top order. The estimate theorems on this page also use different forcing-data hypotheses, as item (iii) records.

(i) Inclusion $C^{2, α} \subseteq W^{2, p}$ for all finite $p$ . If $Ω$ is bounded and $u \in C^{2, α} (\overset{ˉ}{Ω})$ , then each derivative $D^{β} u$ with $∣ β ∣ \leq 2$ is continuous on the compact set $\overset{ˉ}{Ω}$ , and $∥ u ∥_{W^{2, p} (Ω)} \leq ∣ Ω ∣^{1 / p} \sum_{∣ β ∣ \leq 2} \sup_{Ω} ∣ D^{β} u ∣ \leq C (Ω) ∥ u ∥_{C^{2, α} (\overset{ˉ}{Ω})},$ so $C^{2, α} (\overset{ˉ}{Ω}) \subseteq W^{2, p} (Ω)$ with a norm bound depending on the volume and on $p$ through $∣ Ω ∣^{1 / p}$ ; the inclusion is strict, and the Schauder scale is the stronger hypothesis at the top order.
(ii) No reverse inclusion for any finite $p$ . For every $1 < p < \infty$ the space $W^{2, p} (Ω)$ is not contained in $C^{2, α} (\overset{ˉ}{Ω})$ : on the unit ball the function $u (x) = (x_{1})_{+}^{2}$ belongs to $W^{2, p}$ for every finite $p$ while $D_{11} u = 21_{{x_{1} > 0}}$ is discontinuous, as recorded with proof in $W^{2, p}$ regularity implies classical or H"older regularity when $p$ is large. More generally, under the Axiom of Choice, for $n \geq 2$ , and on a bounded $W^{2, p}$ -extension domain, the Sobolev embedding gives at most one H"older derivative, with exponent strictly below $1 - n / p$ when $p > n$ ; finite $p$ never gives the two-derivative H"older estimate.
(iii) The data classes differ in the same direction. The $W^{2, p}$ estimate of Interior $W^{2, p}$ estimate for uniformly elliptic equations with continuous coefficients accepts forcing $L u \in L^{p}$ and concludes an $L^{p}$ bound for $D^{2} u$ , while the Schauder estimate of Interior Schauder estimate for uniformly elliptic equations requires $L u \in C^{0, α}$ and concludes a H"older bound; since $C^{0, α} (\overset{ˉ}{Ω}) \subseteq L^{p} (Ω)$ on a bounded domain with equality false, the Schauder theorem assumes strictly more on the data and concludes strictly more on the solution.
(iv) The endpoints are genuine restrictions of the two theories. The strict range $0 < α < 1$ in the Schauder scale is not a technicality: at the endpoint $α = 1$ the interior estimate fails, and the sharp modulus of $D^{2} N f$ for a Lipschitz source is $∣ x ∣ ∣ \log ∣ x ∣ ∣$ rather than $∣ x ∣$ , with the explicit witness recorded on the companion page (The Schauder estimate fails at the H"older endpoint $α = 1$ ↗). The range $1 < p < \infty$ is the range of the Riesz-multiplier and singular-integral arguments used on this page for the Sobolev estimates; no endpoint $p = 1$ or $p = \infty$ version is asserted here.
(v) Neither scale is a boundary regularity theorem by itself. The $W^{2, p}$ estimate is a priori and assumes $u \in W^{2, p}$ ; a weak solution on a merely Lipschitz domain can fail to reach $W^{2, p}$ altogether at a reentrant corner (Boundary $W^{2, p}$ regularity needs more than Lipschitz boundary ↗), and the radius bookkeeping of both estimates is exercised by The Schauder estimate on a quadratic Poisson solution: radius powers balance ↗.

Thus the two scales should be used according to the data: rough $L^{p}$ forcing is treated by the Sobolev scale at the price of losing H"older regularity at the top order, while $C^{0, α}$ forcing with controlled coefficients is treated by the Schauder scale, which gives two H"older derivatives but no improvement at the endpoint $α = 1$ and no statement for rough coefficients.

The comparison is local in nature: on an infinite-volume domain, boundedness of $u$ and its derivatives does not imply $L^{p}$ integrability: $u \equiv 1$ on $R^{n}$ is a counterexample. An inclusion there requires additional integrability, and on domains with corners both scales require corresponding boundary hypotheses.
The statement of this remark carries no proof obligation of its own: each itemized claim is proved or witnessed in the cited item, and the companion examples page holds the endpoint counterexamples for both scales.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Weak global $W^{2, p}$ regularity for the Dirichlet Laplacian

Statement

Assume the Axiom of Choice and Countable Choice. Let $n \geq 2$ , $n < p < \infty$ , and let $Ω \subset R^{n}$ be a bounded $C^{2, α}$ domain for some $0 < α < 1$ . If $u \in H_{0}^{1} (Ω)$ is a weak solution of $- Δ u = f$ with $f \in L^{p} (Ω)$ , then $u \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ and $∥ u ∥_{W^{2, p} (Ω)} \leq C (∥ f ∥_{L^{p} (Ω)} + ∥ u ∥_{L^{p} (Ω)}),$ where $C = C (n, p, Ω)$ . This is a weak-to-strong regularity theorem; the estimate applies to the weak solution only after its $W^{2, p}$ membership has been established, and the assumption $p > n$ is the range in which the bootstrap of Sobolev exponents terminates at $p$ .

Facts & Assumptions

Given: the Axiom of Choice and Countable Choice, $n \geq 2$ , $n < p < \infty$ , the bounded $C^{2, α}$ domain $Ω$ , $f \in L^{p} (Ω)$ and a weak solution $u \in H_{0}^{1} (Ω)$ of $- Δ u = f$ .

[A1]

Countable Choice is used for the measure-theoretic and Sobolev interfaces; the Axiom of Choice is inherited by the extension, embedding and a priori estimate interfaces and assumed for the quoted solvability input. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Weak formulation (Weak Dirichlet solutions for a divergence-form operator, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field): for the Laplacian the Dirichlet form is $a (v, φ) = \int_{Ω} \nabla v \cdot \nabla φ$ , and $u \in H_{0}^{1} (Ω)$ is a weak solution of $- Δ u = f$ , $f \in L^{2} (Ω)$ , precisely when $\int_{Ω} \nabla u \cdot \nabla φ = \int_{Ω} f φ$ for every $φ \in H_{0}^{1} (Ω)$ . Equivalently, for every $λ \in R$ , $\int_{Ω} \nabla u \cdot \nabla φ + λ \int_{Ω} u φ = \int_{Ω} (f + λ u) φ (φ \in H_{0}^{1} (Ω)) .$ (Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure)

[F2]

Shifted strong solvability (quoted, Haller-Dintelmann Theorem 19.7): let $Ω \subset R^{d}$ be open and bounded with $C^{2}$ boundary and let $L$ have symmetric elliptic principal matrix $a \in C (\overset{ˉ}{Ω})$ , $b, c \in L^{\infty}$ . For each fixed $1 < q < \infty$ there exists $λ_{0} (q) \geq 0$ such that for every $λ \geq λ_{0} (q)$ (indeed $Re λ \geq λ_{0}$ ) and every $h \in L^{q} (Ω)$ the problem $λ z - L z = h$ in $Ω$ , $z = 0$ on $\partial Ω$ , has a unique solution $z \in W^{2, q} (Ω) \cap W_{0}^{1, q} (Ω)$ with $∥ z ∥_{W^{2, q} (Ω)} \leq C (λ, n, q, a, b, c, Ω) ∥ h ∥_{L^{q} (Ω)}$ . For $L = Δ$ ( $a$ the identity matrix, $b = c = 0$ ) this is the solvability of $(λ - Δ) z = h$ in $W^{2, q} \cap W_{0}^{1, q}$ ; the constant may depend on $q$ , and only finitely many exponents $q$ are used below.

[F3]

Higher-order Sobolev embedding (Higher-order Sobolev embedding): on a bounded extension domain, with $k \geq 1$ and $1 \leq q < \infty$ , (a) if $k q < n$ then $W^{k, q} ↪ L^{r}$ for every $1 \leq r \leq n q / (n - k q)$ ; (b) if $k q = n$ then $W^{k, q} ↪ L^{r}$ for every finite $r$ ; (c) if $k q > n$ then $W^{k, q}$ embeds into $L^{\infty}$ and into $C^{0, β} (\overset{ˉ}{Ω})$ for $0 < β < \min {1, k - n / q}$ .

[F4]

$Ω$ is a bounded $C^{k}$ domain for $k = 2$ (Bounded C^k domains and boundary charts), hence a $W^{k, q}$ -extension domain for every $k \leq 2$ and every $1 \leq q \leq \infty$ by Bounded C^k domains admit integer-order Sobolev extension; in particular [F3] applies with $k = 1, 2$ and all exponents used below. (Sobolev extension domains and extension operators)

[F5]

A priori estimate (Global $W^{2, p}$ Dirichlet estimate on a $C^{1, 1}$ domain): for the bounded $C^{1, 1}$ domain $Ω$ (a $C^{2, α}$ domain is $C^{1, 1}$ ) there is $C_{0} = C_{0} (n, p, Ω)$ with $∥ v ∥_{W^{2, p} (Ω)} \leq C_{0} (∥ Δ v ∥_{L^{p} (Ω)} + ∥ v ∥_{L^{p} (Ω)})$ for every $v \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ .

[F6]

Energy uniqueness (Weak Dirichlet solutions for a divergence-form operator): if $w \in H_{0}^{1} (Ω)$ satisfies $\int_{Ω} \nabla w \cdot \nabla φ + λ \int_{Ω} w φ = 0$ for every $φ \in H_{0}^{1} (Ω)$ and $λ > 0$ , then $w = 0$ : testing with $φ = \overline{w}$ (or $w$ for real scalars) gives $\int ∣ \nabla w ∣^{2} + λ \int ∣ w ∣^{2} = 0$ .

[F7]

Under Countable Choice, the map from $L_{l o c}^{1} (Ω)$ classes to distributions given by $g \mapsto (φ \mapsto \int_{Ω} g φ)$ is injective; equal regular distributions therefore come from functions equal almost everywhere (Locally integrable functions embed in distributions).

Proof

technique · direct

1.1F3F4algebraA1

Initial integrability and the exponent list. Since $u \in H_{0}^{1} (Ω) = W_{0}^{1, 2} (Ω)$ and $Ω$ is a bounded extension domain for $W^{1, 2}$ by [F4], the embedding [F3] with $k = 1$ , $q = 2$ gives $u \in L^{r} (Ω)$ for every $2 \leq r \leq \frac{2 n}{n - 2}$ if $n > 2$ , and for every finite $r$ if $n = 2$ . Put $q_{0} : = \min {p, \frac{2 n}{n - 2}} (n > 2), q_{0} : = p (n = 2),$ so that $2 \leq q_{0} \leq p$ and $u \in L^{q_{0}} (Ω)$ . If $q_{0} = p$ , take the exponent list to be the singleton $Q = {p}$ and set $m = 0$ . Otherwise define a strictly increasing finite list $q_{0} < q_{1} < \dots < q_{m} = p$ by $q_{i + 1} : = \min {p, \frac{n q_{i}}{n - 2 q_{i}}}$ when $2 q_{i} < n$ and $q_{i + 1} : = p$ when $2 q_{i} \geq n$ . The list is finite and depends only on $n, p$ : while $q_{i} < p$ and $2 q_{i} < n$ one has $\frac{1}{q_{i + 1}} = \frac{1}{q_{i}} - \frac{2}{n}$ (unless the minimum is $p$ , which ends the list), so the reciprocals decrease by the fixed positive amount $\frac{2}{n}$ and the process reaches either $p$ or the region $2 q_{i} \geq n$ after at most $⌈ \frac{n}{2} (\frac{1}{q_{0}} - \frac{1}{p}) ⌉$ steps, after which it reaches $p$ in one more step.

2.1F1F2F6step 1.1algebra

Choice of the shift and the first solve. Let $Q : = {q_{i} : 0 \leq i \leq m}$ be the finite set of exponents in the list, so $p \in Q$ and this definition also covers the case $q_{0} = p$ (then $m = 0$ and $Q = {p}$ ). For each $q \in Q$ , apply [F2] to $L = Δ$ and let $λ_{0} (q)$ be its threshold; choose $λ > \max ({0} \cup {λ_{0} (q) : q \in Q})$ . Since $f \in L^{p} (Ω) \subseteq L^{q_{0}} (Ω)$ and $u \in L^{q_{0}} (Ω)$ by step 1.1, the datum $h_{0} : = f + λ u$ lies in $L^{q_{0}} (Ω)$ ; by [F2] there is $z_{0} \in W^{2, q_{0}} (Ω) \cap W_{0}^{1, q_{0}} (Ω)$ with $(λ - Δ) z_{0} = h_{0}$ strongly, hence weakly by [F1] (test against compactly supported smooth functions and use density). The weak solution $u$ satisfies the same shifted weak equation with datum $h_{0}$ , as recorded in [F1]. Since $q_{0} \geq 2$ , the space $W_{0}^{1, q_{0}} (Ω)$ is contained in $H_{0}^{1} (Ω)$ (bounded $Ω$ gives $W^{1, q_{0}} (Ω) \subseteq W^{1, 2} (Ω)$ and the closures transfer), so $z_{0} - u \in H_{0}^{1} (Ω)$ and [F6] gives $z_{0} = u$ . Hence $u \in W^{2, q_{0}} (Ω) \cap W_{0}^{1, q_{0}} (Ω)$ .

3.1F1F2F3F6step 2.1induction

The bootstrap induction. Suppose $u \in W^{2, q_{i}} (Ω) \cap W_{0}^{1, q_{i}} (Ω)$ with $q_{i} < p$ . If $2 q_{i} < n$ , then F3 with $k = 2$ , $q = q_{i}$ gives $u \in L^{r}$ for every $r \leq \frac{n q_{i}}{n - 2 q_{i}}$ , in particular $u \in L^{q_{i + 1}}$ ; if $2 q_{i} = n$ , then F3 gives $u \in L^{r}$ for every finite $r$ , so $u \in L^{p} = L^{q_{i + 1}}$ ; if $2 q_{i} > n$ , then F3 gives $u \in L^{\infty} \subseteq L^{p} = L^{q_{i + 1}}$ . In all three cases $h_{i} : = f + λ u \in L^{q_{i + 1}}$ because $f \in L^{p}$ and $q_{i + 1} \leq p$ ; by [F2] applied at the exponent $q_{i + 1}$ there is $z_{i + 1} \in W^{2, q_{i + 1}} (Ω) \cap W_{0}^{1, q_{i + 1}} (Ω)$ solving $(λ - Δ) z_{i + 1} = h_{i}$ ; by [F1] and [F6], applied exactly as in step 2.1 (with $q_{i + 1} \geq q_{i} \geq 2$ so that $z_{i + 1} - u \in H_{0}^{1}$ ), we get $z_{i + 1} = u$ and hence $u \in W^{2, q_{i + 1}}$ . Induction over the finite list gives $u \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ .

4.1step 3.1F1F5F7algebra

The estimate and almost-everywhere equation. Now that $u \in W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ , [F5] applies with $v = u$ : $∥ u ∥_{W^{2, p}} \leq C_{0} (∥ Δ u ∥_{L^{p}} + ∥ u ∥_{L^{p}})$ . For every $φ \in C_{c}^{\infty} (Ω)$ , the weak equation [F1] and the definition of the weak Laplacian give $\int_{Ω} (- Δ u) φ = \int_{Ω} \nabla u \cdot \nabla φ = \int_{Ω} f φ .$ Both $- Δ u$ and $f$ lie in $L^{p} (Ω) \subseteq L_{l o c}^{1} (Ω)$ , so their regular distributions agree; injectivity [F7] gives $- Δ u = f$ almost everywhere. Hence $∥ Δ u ∥_{L^{p}} = ∥ f ∥_{L^{p}}$ and the displayed bound holds with $C = C_{0}$ .

5.1step 3.1step 4.1F2F5given∎

Conclusion. The weak solution $u \in H_{0}^{1} (Ω)$ of $- Δ u = f$ with $f \in L^{p} (Ω)$ , $p > n$ , is shown to lie in $W^{2, p} (Ω) \cap W_{0}^{1, p} (Ω)$ , and the a priori estimate of [F5] then gives $∥ u ∥_{W^{2, p}} \leq C (∥ f ∥_{L^{p}} + ∥ u ∥_{L^{p}})$ with $C$ depending only on $n, p, Ω$ . The shifted-equation argument uses the finite sequence of Sobolev exponents and the unique solvability [F2] at each of them; it never assumes $W^{2, p}$ regularity of $u$ in advance.

Remarks

The proof shows precisely how the range $p > n$ is used: the weak solution starts in $H_{0}^{1}$ , the shifted strong solvability lifts one Sobolev order at a time, and the higher-order embedding converts a $W^{2, q}$ bound into a higher $L^{r}$ bound; reciprocals decrease by $2 / n$ per step, so the process reaches any prescribed finite exponent after finitely many steps.
The bridge from the literature's strong solvability theorem to the given weak solution is the shifted equation and energy uniqueness [F6], not an assumption of $W^{2, p}$ regularity. No maximum principle, no symmety of the domain and no spectral theory beyond the threshold $λ_{0}$ of [F2] is used.
The domain is assumed $C^{2, α}$ for some $α \in (0, 1)$ , which is stronger than the $C^{1, 1}$ of the a priori estimate [F5] and is used only through the $C^{2}$ extension and boundary requirements of [F2] and [F3].

5 · Examples, counterexamples and false statements

None yet.

Schauder and LpLp Elliptic Estimates

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Hölder spaces Ck,αCk,α, closure and interior scaled norms, and Ck,αCk,α domains

Definition

Remarks

The closure Hölder spaces are Banach spaces

Statement

Facts & Assumptions

Proof

Remarks

The cancelled representation of the second derivatives of Newtonian potentials

Statement

Facts & Assumptions

Proof

Remarks

Uniformly elliptic nondivergence-form operators and their frozen coefficients

Definition

Remarks

Ehrling-type Hölder and derivative interpolation with an epsilon loss

Statement

Facts & Assumptions

Proof

Remarks

Freezing coefficients makes the Schauder error absorbable on a small ball

Statement

Facts & Assumptions

Proof

Remarks

Interior Schauder estimate for uniformly elliptic equations

Statement

Facts & Assumptions

Proof

Remarks

C2,αC2,α boundary flattening preserves the nondivergence structure, ellipticity and Hölder norms

Statement

Facts & Assumptions

Proof

Remarks

Boundary Schauder estimate for the Dirichlet problem

Statement

Facts & Assumptions

Proof

Remarks

The method of continuity for a uniformly estimated affine family of bounded operators

Statement

Facts & Assumptions

Proof

Remarks

Global W2,pW2,p estimate for the Laplacian on Euclidean space

Statement

Facts & Assumptions

Proof

Remarks

LpLp interpolation absorption of first derivatives by second derivatives

Statement

Facts & Assumptions

Proof

Remarks

The cutoff commutator in the local W2,pW2,p estimates

Statement

Facts & Assumptions

Proof

Remarks

Local W2,pW2,p regularity of weak solutions of the Poisson equation

Statement

Facts & Assumptions

Proof

Remarks

Interior W2,pW2,p estimate for uniformly elliptic equations with continuous coefficients

Statement

Facts & Assumptions

Proof

Remarks

Global W2,pW2,p Dirichlet estimate on a C1,1C1,1 domain

Statement

Facts & Assumptions

Proof

Schauder and $L^{p}$ Elliptic Estimates

Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains

$C^{2, α}$ boundary flattening preserves the nondivergence structure, ellipticity and Hölder norms

Global $W^{2, p}$ estimate for the Laplacian on Euclidean space

$L^{p}$ interpolation absorption of first derivatives by second derivatives

The cutoff commutator in the local $W^{2, p}$ estimates

Local $W^{2, p}$ regularity of weak solutions of the Poisson equation

Interior $W^{2, p}$ estimate for uniformly elliptic equations with continuous coefficients

Global $W^{2, p}$ Dirichlet estimate on a $C^{1, 1}$ domain

Injectivity removes the $L^{p}$ term from the global $W^{2, p}$ estimate

$W^{2, p}$ regularity implies classical or H"older regularity when $p$ is large

The Schauder and $W^{2, p}$ scales are different, not interchangeable

Weak global $W^{2, p}$ regularity for the Dirichlet Laplacian