Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.10349
- arXiv:
- arXiv:2405.10349
- Bibcode:
- 2024arXiv240510349G
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35A23;
- 26D10;
- 35Q74/35Q75;
- 46E35