Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition
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| Format: | Preprint |
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2021
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| author | Doležalová, Anna Hencl, Stanislav Malý, Jan |
| author_facet | Doležalová, Anna Hencl, Stanislav Malý, Jan |
| contents | Let $Ω,Ω'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:Ω\toΩ'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_Ω(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak limit of $f_m$ in $W^{1,2}$. We show that $f$ is invertible a.e., more precisely it satisfies the (INV) condition of Conti and De Lellis and thus it has all the nice properties of mappings in this class.
Generalization to higher dimensions and an example showing sharpness of the condition $1/J^2_f\in L^1$ are also given. Using this example we also show that unlike the planar case the class of weak limits and the class of strong limits of $W^{1,2}$ Sobolev homeomorphisms in $\mathbb{R}^3$ are not the same. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_08041 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition Doležalová, Anna Hencl, Stanislav Malý, Jan Functional Analysis Let $Ω,Ω'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:Ω\toΩ'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_Ω(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak limit of $f_m$ in $W^{1,2}$. We show that $f$ is invertible a.e., more precisely it satisfies the (INV) condition of Conti and De Lellis and thus it has all the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition $1/J^2_f\in L^1$ are also given. Using this example we also show that unlike the planar case the class of weak limits and the class of strong limits of $W^{1,2}$ Sobolev homeomorphisms in $\mathbb{R}^3$ are not the same. |
| title | Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2112.08041 |