False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

Exterior spheres and continuous boundary data do not give global bounded C2alpha solvability

Statement

Let $0 < α < 1$ and let $U \subset R^{d}$ be bounded and satisfy an exterior sphere condition at every boundary point. Let $a_{i j} = a_{j i} \in C^{1, α} (U)$ be uniformly elliptic with minimum eigenvalue at least $λ_{0} > 0$ , let $b_{i} \in C^{1, α} (U)$ and $c_{i}, d \in C^{α} (U)$ , and define $L [u] = \sum_{i, j = 1}^{d} D_{i} (a_{i j} D_{j} u + b_{i} u) + \sum_{i = 1}^{d} c_{i} D_{i} u + d u .$ Assume $\sum_{i, j} ∣ a_{i j} (x) ∣^{2} \leq Λ$ and $λ_{0}^{- 2} (∥ b (x) ∥^{2} + ∥ c (x) ∥^{2}) + λ_{0}^{- 1} ∣ d (x) ∣ \leq v^{2}$ on $U$ , and use the expanded coefficients ${\tilde{b}}_{j} = \sum_{i} D_{i} a_{i j} + b_{j} + c_{j}$ and $\tilde{c} = \sum_{i} D_{i} b_{i} + d \leq 0$ . For every $f \in C^{0, α} (U)$ and $g \in C^{0} (\partial U)$ , the source assertion is that the Dirichlet problem $L [u] = f$ in $U$ , $u = g$ on $\partial U$ has a unique solution $u \in C^{2, α} (U) \cap C^{0} (\overset{ˉ}{U})$ . For any $h \in C^{0} (U)$ with $L [h] \in C^{0, α} (U)$ and $h \in C^{0, α} (\partial U)$ , the source further asserts $∥ u - h ∥_{C^{0} (U)} \leq C (∥ f - L [h] ∥_{C^{0} (U)} + ∥ g - h ∥_{C^{0} (\partial U)}),$ where $C$ depends only on the domain and the displayed coefficient bounds. The global existence part of this assertion is false under the library's bounded global Hölder convention.

Statement references

Shin et al., Theorem D.1 and Lemma D.2, PDF p.26: source assertion retained here

Exterior-sphere domains need not have global C2alpha Dirichlet solutions

Facts & Assumptions

Given: The source assertion in the Statement, including its printed divergence indexing. The refutation uses $b = c = d = 0$ , so the indexing discrepancy does not affect it.

[A1]

Assume for contradiction that the asserted global existence conclusion holds for every admissible datum.

[F1]

The sector $U = {0 < r < 1, 0 < θ < 2 π / 3}$ is bounded Lipschitz and satisfies the exterior-sphere condition. With $L = Δ$ , $g = 0$ and $u = χ (r) r^{3 / 2} \sin (3 θ / 2)$ , its smooth forcing $f = Δ u$ is globally Hölder, and $u$ is classical and continuous on the closure, but has unbounded second derivatives (Exterior-sphere domains need not have global C2alpha Dirichlet solutions).

[F2]

The zero-boundary classical homogeneous Dirichlet problem is unique by comparison (Elliptic residual and boundary stability from a maximum principle).

[F3]

The global bounded Hölder norm includes the supremum of every second derivative (Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains). The positive-principal convention is fixed by Shin's positive divergence-form operator.

Refutation

1.1assume-contraF1F3

Assume for contradiction the claimed global existence assertion [A1]. Use the sector and data from [F1]. The coefficient matrix is the identity and all lower coefficients vanish; uniform ellipticity holds with $λ_{0} = 1$ , the pointwise source bounds hold with $Λ = 2$ and $v = 0$ , and the expanded zero-order coefficient is zero. Thus these data satisfy every hypothesis of the claimed existence assertion.

1.2A1F1F2

Assumption [A1] gives a global bounded $C^{2, α} (U)$ solution $u^{*}$ continuous on the closure. The explicit $u$ from [F1] is a second classical continuous Dirichlet solution. Their difference is zero-boundary harmonic, so [F2] implies $u^{*} = u$ .

2.1step 1.2F1F3discharge-contradiction∎

Along $θ = π / 3$ and $r < 1 / 4$ , $\partial_{r r} u = (3 / 4) r^{- 1 / 2}$ is unbounded. A globally bounded Cartesian Hessian would bound this unit-direction second derivative. Therefore [F3] excludes $u$ from the asserted global class, contradicting step 1.2. The global existence claim is false; its failure does not refute the separately proved classical residual estimate.

Remarks

The printed source formula also repeats $b_{i} u$ inside the double divergence sum, although its stated expanded drift contains $b$ only once. This defect is retained visibly in the historical assertion above. The native Laplacian sector refutation is independent of that discrepancy. The repaired existence theorem uses the intended single $b$ divergence and preserves the correct local-interior and conditional smooth-boundary branches.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shin, Darbon, and Karniadakis, On the Convergence of Physics-Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs (standard reference, not scraped)