Continuity for Sobolev mappings with null Lagrangian bounds
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912684654460928 |
|---|---|
| author | Kangasniemi, Ilmari Onninen, Jani |
| author_facet | Kangasniemi, Ilmari Onninen, Jani |
| contents | We prove the continuity of Sobolev functions $φ\in W^{1,n}_{\mathrm{loc}}(Ω)$, $Ω\subset \mathbb{R}^n$, that satisfy \[
\lvert\nabla φ(x)\rvert^n \le K(x)\bigl(\langle \nabla φ(x), ξ(x)\rangle + A(x)\bigr), \] where $ξ\in L_{\mathrm{loc}}^{n/(n-1)}(Ω, \mathbb{R}^n)$ is weakly divergence-free, and $K \in L^p_{\mathrm{loc}} (Ω)$, $A \in L^q_{\mathrm{loc}} (Ω)$ are non-negative with $p^{-1}+q^{-1}<1$. The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for $f \in W^{1,n}_{\mathrm{loc}} (Ω, \mathbb{R}^n)$ satisfying the distortion inequality with defect $\lvert Df(x)\rvert^n \le K(x)\det Df(x) + Σ(x)$; this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuity for Sobolev mappings with null Lagrangian bounds Kangasniemi, Ilmari Onninen, Jani Complex Variables Analysis of PDEs Classical Analysis and ODEs Functional Analysis 30C65 (Primary) 35R45, 46E35 (Secondary) We prove the continuity of Sobolev functions $φ\in W^{1,n}_{\mathrm{loc}}(Ω)$, $Ω\subset \mathbb{R}^n$, that satisfy \[ \lvert\nabla φ(x)\rvert^n \le K(x)\bigl(\langle \nabla φ(x), ξ(x)\rangle + A(x)\bigr), \] where $ξ\in L_{\mathrm{loc}}^{n/(n-1)}(Ω, \mathbb{R}^n)$ is weakly divergence-free, and $K \in L^p_{\mathrm{loc}} (Ω)$, $A \in L^q_{\mathrm{loc}} (Ω)$ are non-negative with $p^{-1}+q^{-1}<1$. The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for $f \in W^{1,n}_{\mathrm{loc}} (Ω, \mathbb{R}^n)$ satisfying the distortion inequality with defect $\lvert Df(x)\rvert^n \le K(x)\det Df(x) + Σ(x)$; this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets. |
| title | Continuity for Sobolev mappings with null Lagrangian bounds |
| topic | Complex Variables Analysis of PDEs Classical Analysis and ODEs Functional Analysis 30C65 (Primary) 35R45, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2509.20326 |