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Math-2861 · mathematics · Pde

Strong diffeomorphic approximation of three-dimensional Sobolev homeomorphisms up to exponent two

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For arbitrary bounded connected open domains $Omega,Lambda\subset R^3$, every $1\leq p\leq 2$ and every $W^{1,p}$ homeomorphism f:Omega$\to$Lambda, do smooth $W^{1,p}$ diffeomorphisms $f_j$ between the same domains exist with $integral_Omega(|f_j-f|^p+|Df_j-Df|^p)\to 0$? No inverse Sobolev estimate or quasiconformal bound is requested.

Short English statement

For arbitrary bounded connected open domains $Omega,Lambda\subset R^3$, every $1\leq p\leq 2$ and every $W^{1,p}$ homeomorphism f:Omega$\to$Lambda, do smooth $W^{1,p}$ diffeomorphisms $f_j$ between the same domains exist with $integral_Omega(|f_j-f|^p+|Df_j-Df|^p)\to 0$? No inverse Sobolev estimate or quasiconformal bound is requested.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2861-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2861

Ball--Evans approximation for low exponents: The complementary range includes both endpoints $p=1$ and $p=2$. The approximation preserves the source and target domains and converges in function and gradient norms globally, without imposing inverse Sobolev regularity or quasiconformal distortion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Strong diffeomorphic approximation of three-dimensional Sobolev homeomorphisms up to exponent two

OpenAI's claim

Ball--Evans approximation for low exponents: For $1\le p\le2$, the manuscript claims strong $W^{1,p}$ approximation of every three-dimensional Sobolev homeomorphism between bounded domains by smooth diffeomorphisms onto the same target. Arbitrary domain boundaries are allowed.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\Omega,\Lambda\subset\mathbb R^3$ be nonempty bounded connected open sets and let $1\le p\le2$. For every homeomorphism $f:\Omega\to\Lambda$ in $W^{1,p}(\Omega;\mathbb R^3)$, the claim provides smooth diffeomorphisms $f_j:\Omega\to\Lambda$, each in the same Sobolev class, such that \[\int_\Omega(|f_j-f|^p+|Df_j-Df|^p)\,dx\longrightarrow0.\] The source and target are preserved exactly; no boundary regularity, prescribed boundary values or convergence of inverse derivatives is assumed.

Outline of the argument

Ball--Evans approximation for low exponents: Source curves and, at the critical exponent, surfaces are sampled against complementary dual pieces of a target triangulation. Their crossing counts become length and area budgets after local topological changes and target reparametrization. A locally bi-Lipschitz replacement preserves buffered data, and summable local smoothing errors control the full forward Sobolev norm, including the $p=1$ endpoint.

Formal verification

Ball--Evans approximation for low exponents: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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