Constant rank operators in Korn-Maxwell-Sobolev inequalities

Abstract

We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form \[ \|P - Π_{\mathbb{B}} Π_{\ker\mathscr{A}} P\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} \le c \, (\|\mathscr{A}[P]\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} + \|\mathbb{B} P\|_{\mathrm{L}^p(\mathbb{R}^n)}) \] for all $P\in {\mathrm{C}}_c^{\infty }({\mathbb{R}}^n;V)$, where $V$ is a finite-dimensional vector space, $\mathcal{A}$ is a linear mapping, and $\mathbb{B}$ is a constant coefficient homogeneous differential operator of order $k$. In particular, we can treat the combination $(p,\mathcal{A},\mathbb{B},k)=(1,\mathrm{tr},\mathrm{Curl},1)$. Our results generalize the techniques from Gmeineder et al. (Math.Mod.Met.Appl.Sci,34:03,2024; arXiv:2405.10349), which exclusively dealt with reduced elliptic operators. In contrast to the reduced ellipticity case, however, the reduced constant rank case necessitates to introduce a correction, namely the projection ${\Pi }_{\mathbb{B}}$ on the left-hand side of the inequality.

arXiv comment7 pages, comments welcome