Note on injectivity in second-gradient Nonlinear Elasticity
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915373840859136 |
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| author | Hencl, Stanislav Mohanta, Kaushik |
| author_facet | Hencl, Stanislav Mohanta, Kaushik |
| contents | Let $q>1$, $(1-\frac{1}{q})a\geq 1$ and let $Ω\subset \mathbb{R}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(Ω,\mathbb{R}^2)$ with $|J_f|^{-a}\in L^1(Ω)$ are homeomorphism if they agree with a homeomorphism on the boundary. The condition $(1-\frac{1}{q})a\geq 1$ is sharp. We also have a new sharp result about the $\mathcal{H}^{n-1}$ measure of the projection of the set $\{J_f=0\}$ in $\mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04938 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note on injectivity in second-gradient Nonlinear Elasticity Hencl, Stanislav Mohanta, Kaushik Analysis of PDEs Functional Analysis Let $q>1$, $(1-\frac{1}{q})a\geq 1$ and let $Ω\subset \mathbb{R}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(Ω,\mathbb{R}^2)$ with $|J_f|^{-a}\in L^1(Ω)$ are homeomorphism if they agree with a homeomorphism on the boundary. The condition $(1-\frac{1}{q})a\geq 1$ is sharp. We also have a new sharp result about the $\mathcal{H}^{n-1}$ measure of the projection of the set $\{J_f=0\}$ in $\mathbb{R}^n$. |
| title | Note on injectivity in second-gradient Nonlinear Elasticity |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2507.04938 |