preprint

Inserted: 10 jan 2025
Last Updated: 10 jan 2025

Year: 2024

Abstract:

We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form \[ \lVert P - \Pi_{\mathbb{B}} \Pi_{\ker\mathscr{A}} P\rVert_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} \le c \, ( \lVert\mathscr{A}[P] \rVert_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} + \lVert\mathbb{B} P \rVert_{\mathrm{L}^p(\mathbb{R}^n)}) \] for all $ P \in \mathrm{C}_c^\infty(\mathbb{R}^n; V) $, where $ V $ is a finite-dimensional vector space, $ \mathscr{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 $(p,\mathscr{A},\mathbb{B},k)=(1,\operatorname{tr},\operatorname{Curl},1)$. 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.