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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.

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Sources

Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025; complete 281-page graduate lecture notes) (standard reference, not scraped)
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)