LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Linear fourth derivative bound with controlled realized coefficients

Statement

Work in ZFC. Let $n \geq 2$ be an integer, $0 < α < 1$ , and let $A^{'}, A \subset R^{n}$ be fixed bounded Euclidean annuli with $\overline{A^{'}} \subset A$ . Let $z$ be smooth on $A$ and solve $L z = a^{i j} z_{i j} + b^{j} z_{j} + c z = f$ , with summation over $i, j = 1, \dots, n$ . Suppose $a$ is symmetric and $λ ∣ ξ ∣^{2} \leq a^{i j} (x) ξ_{i} ξ_{j} \leq Λ ∣ ξ ∣^{2}$ for all $x \in A$ and $ξ \in R^{n}$ , where $0 < λ \leq Λ < \infty$ . Suppose every component of $a, b, c$ has $C^{2, α} (A)$ norm at most $M < \infty$ , and $f \in C^{2, α} (A)$ . Then $∥ z ∥_{C^{4, α} (A^{'})} \leq C (∥ z ∥_{L^{\infty} (A)} + ∥ f ∥_{C^{2, α} (A)}) .$ Here $C$ depends only on $n, α, λ, Λ, M$ and the fixed annular geometry, including the positive separation $dist (\overline{A^{'}}, \partial A)$ . The coefficients are the realized functions of position, even if originally obtained by evaluating a nonlinear operator at $z, D z$ ; their stated bounds must hold as functions of position. In particular the same $C$ works for any family with these common bounds.

Facts & Assumptions

[F1]

The interior linear Schauder estimate applies on a ball to a bounded classical solution of a uniformly elliptic equation with controlled $C^{0, α}$ coefficients, including first-order and zeroth-order terms. Its constant depends only on dimension, exponent, ellipticity and the stated scaled coefficient bounds. Interior Schauder estimate for uniformly elliptic equations

[F2]

Use the Euclidean Hölder norms and their scaled ball versions. The product estimate needed below follows directly from $p (x) q (x) - p (y) q (y) = p (x) (q (x) - q (y)) + q (y) (p (x) - p (y))$ : $[p q]_{0, α} \leq ∥ p ∥_{\infty} [q]_{0, α} + ∥ q ∥_{\infty} [p]_{0, α}$ . Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

[F3]

The repeated-index expression $L = a^{i j} \partial_{i} \partial_{j} + b^{j} \partial_{j} + c$ and the constants $λ, Λ$ have the uniformly elliptic nondivergence-operator convention. Differentiating the equation keeps this same principal operator. Uniformly elliptic nondivergence-form operators and their frozen coefficients

[A1]

Work in ZFC. Full Choice implies the Countable Choice assumed by the Schauder supplier; that implication is the use of Choice in each invocation of [F1]. The Axiom of Choice

Proof

Given: The hypotheses in the Statement.

Goal: Establish the asserted estimate with the stated uniform dependence.

1.1F1F2F3A1givenalgebra

If $∥ z ∥_{L^{\infty} (A)}$ is infinite, the claimed inequality is automatic in the extended sense, so assume it finite and put $N = ∥ z ∥_{L^{\infty} (A)} + ∥ f ∥_{C^{2, α} (A)}$ . Fix intermediate open sets with $A^{'} ⋐ U_{2} ⋐ U_{1} ⋐ A$ , where $V ⋐ W$ means $\overline{V}$ is compact and contained in $W$ . For any of these three successive inclusions, positive separation allows balls of a common fixed radius centered in the smaller set and contained in the larger set. Apply [F1] on these balls and convert its scaled norms to ordinary norms. The local derivative bounds give the global suprema on the smaller set. For Hölder quotients, pairs closer than half a target-ball radius lie in one target ball; separated pairs are bounded by twice the derivative supremum divided by that fixed distance to the power $α$ . A finite cover of the compact closure gives the same bounds, with constants depending only on the fixed geometry. The coefficient $C^{0, α}$ bounds needed on each ball follow from the assumed $C^{2, α}$ bounds and the derivative bounds on that convex ball. Consequently [F1] applied to $L z = f$ gives $∥ z ∥_{C^{2, α} (U_{1})} \leq C N$ .

2.1step 1.1F1F2F3A1givenalgebra

Differentiate once in coordinate $k$ . The equation is $L z_{k} = f_{k} - a_{k}^{i j} z_{i j} - b_{k}^{j} z_{j} - c_{k} z$ . On the balls contained in $U_{1}$ , its right-hand side has $C^{0, α}$ norm at most $C N$ by step 1.1, the coefficient bounds and the product estimate [F2]. Also $∥ z_{k} ∥_{L^{\infty} (U_{1})} \leq C N$ . Applying [F1] with the same elliptic operator and assembling the ball estimates as in step 1.1 yields $∥ z ∥_{C^{3, α} (U_{2})} \leq C N$ .

3.1step 1.1step 2.1F1F2F3A1givenalgebra∎

Differentiate twice. The equation for $z_{k ℓ}$ has the same left-hand operator $L$ and right-hand side $f_{k ℓ} - a_{k ℓ}^{i j} z_{i j} - a_{k}^{i j} z_{i j ℓ} - a_{ℓ}^{i j} z_{i j k} - b_{k ℓ}^{j} z_{j} - b_{k}^{j} z_{j ℓ} - b_{ℓ}^{j} z_{j k} - c_{k ℓ} z - c_{k} z_{ℓ} - c_{ℓ} z_{k}$ . On balls contained in $U_{2}$ its $C^{0, α}$ norm is at most $C N$ by step 2.1, [F2] and the coefficient bounds; the same bound holds for $∥ z_{k ℓ} ∥_{\infty}$ . Apply [F1] to every $z_{k ℓ}$ and assemble the estimates on $A^{'}$ by step 1.1. This bounds $z$ in $C^{4, α} (A^{'})$ by $C N$ . All constants depend only on the listed geometry, dimension, exponent, ellipticity and realized coefficient bounds, so the asserted family uniformity follows.

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.