Weak limits of Sobolev homeomorphisms are one to one
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909303096475648 |
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| author | Bouchala, Ondřej Hencl, Stanislav Zhu, Zheng |
| author_facet | Bouchala, Ondřej Hencl, Stanislav Zhu, Zheng |
| contents | We prove that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions.
Let $Ω\subseteq \mathbb{R}^n$ be a domain and let $p>\left\lfloor\frac{n}{2}\right\rfloor$ for $n\geq 4$ or $p\geq 1$ for $n=2,3$. Assume that $f_k\in W^{1,p}$ is a sequence of homeomorphisms such that $f_k\rightharpoonup f$ weakly in $W^{1,p}$ and assume that $J_f>0$ a.e. Then we show that $f$ is injective a.e. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01260 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak limits of Sobolev homeomorphisms are one to one Bouchala, Ondřej Hencl, Stanislav Zhu, Zheng Functional Analysis We prove that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions. Let $Ω\subseteq \mathbb{R}^n$ be a domain and let $p>\left\lfloor\frac{n}{2}\right\rfloor$ for $n\geq 4$ or $p\geq 1$ for $n=2,3$. Assume that $f_k\in W^{1,p}$ is a sequence of homeomorphisms such that $f_k\rightharpoonup f$ weakly in $W^{1,p}$ and assume that $J_f>0$ a.e. Then we show that $f$ is injective a.e. |
| title | Weak limits of Sobolev homeomorphisms are one to one |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2409.01260 |