Abstract:Let $X, Y \subset \mathbb{R}^n$ be Lipschitz domains, and suppose there is a homeomorphism $\varphi \colon \overline{X} \to \overline{Y}$. We consider the class of Sobolev mappings $f \in W^{1,n} (X, \mathbb{R}^n)$ with a strictly positive Jacobian determinant almost everywhere, whose Sobolev trace coincides with $\varphi$ on $\partial X$. We prove that every mapping in this class extends continuously to $\overline{X}$ and is a monotone (continuous) surjection from $\overline{X}$ onto $\overline{Y}$ in the sense of C.B. Morrey. As monotone mappings, they may squeeze but not fold the reference configuration $X$. This behavior reflects weak interpenetration of matter, as opposed to folding, which corresponds to strong interpenetration. Despite allowing weak interpenetration of matter, these maps are globally invertible, generalizing the pioneering work of J.M. Ball.
| Subjects: | Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA) |
| MSC classes: | 46E35 |
| Cite as: | arXiv:2507.07206 [math.AP] |
| (or arXiv:2507.07206v2 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2507.07206 arXiv-issued DOI via DataCite |
From: Sabrina Traver [view email]
[v1]
Wed, 9 Jul 2025 18:29:55 UTC (14 KB)
[v2]
Fri, 22 May 2026 02:09:33 UTC (13 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。