Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910450353963008 |
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| author | Gmeineder, Franz Lewintan, Peter Van Schaftingen, Jean |
| author_facet | Gmeineder, Franz Lewintan, Peter Van Schaftingen, Jean |
| contents | We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities \begin{align*}
\lVert P\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}\le c\big(\lVert\mathscr{A}[P]\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}+\lVert\mathbb{B}P\rVert_{L^{1}(\mathbb{R}^n)}\big)
\end{align*} to hold for all $P\in C_{c}^{\infty}(\mathbb{R}^{n};V)$, where $\mathscr{A}$ is a linear map between finite dimensional vector spaces and $\mathbb{B}$ is a $k$-th order, linear and homogeneous constant-coefficient differential operator. By the appearance of the $L^{1}$-norm of the differential expression $\mathbb{B}P$ on the right-hand side, such inequalities generalise previously known estimates to the borderline case $p=1$, and thereby answer an open problem due to Müller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10349 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities Gmeineder, Franz Lewintan, Peter Van Schaftingen, Jean Analysis of PDEs 35A23, 26D10, 35Q74/35Q75, 46E35 We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities \begin{align*} \lVert P\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}\le c\big(\lVert\mathscr{A}[P]\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}+\lVert\mathbb{B}P\rVert_{L^{1}(\mathbb{R}^n)}\big) \end{align*} to hold for all $P\in C_{c}^{\infty}(\mathbb{R}^{n};V)$, where $\mathscr{A}$ is a linear map between finite dimensional vector spaces and $\mathbb{B}$ is a $k$-th order, linear and homogeneous constant-coefficient differential operator. By the appearance of the $L^{1}$-norm of the differential expression $\mathbb{B}P$ on the right-hand side, such inequalities generalise previously known estimates to the borderline case $p=1$, and thereby answer an open problem due to Müller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative. |
| title | Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities |
| topic | Analysis of PDEs 35A23, 26D10, 35Q74/35Q75, 46E35 |
| url | https://arxiv.org/abs/2405.10349 |