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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.

Depends on

Used by

Injectivity removes the Lᵖ term from the global W^2,p estimate Corollary
Boundary W^2,p regularity needs more than Lipschitz boundary Counterexample
Weak global W^2,p regularity for the Dirichlet Laplacian Theorem

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Sources

John Villavert, Elementary Theory and Methods for Elliptic Partial Differential Equations (2017; complete 220-page lecture notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford University; complete 118-page author notes, Chapter 12 Schauder Theory) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)

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