KMS Inequalities: From Elliptic Operators to Constant Rank

TL;DR

This work extends Korn-Maxwell-Sobolev (KMS) inequalities from elliptic operators to operators with constant rank on the kernel of a given differential constraint $\mathscr{A}$. The main contribution is a left-hand side correction term $\Pi_{\mathbb{B}} \mathcal{P}_{\ker(\mathscr{A})}[P]$, yielding $\| P - \Pi_{\mathbb{B}} \mathcal{P}_{\ker(\mathscr{A})}[P] \|_{\dot W^{k-1,p^*}} \lesssim \| \mathscr{A}[P] \|_{\dot W^{k-1,p^*}} + \| \mathbb{B}P \|_{L^p}$ for $1<p<n$, provided $\mathbb{B}$ has constant rank on $\ker(\mathscr{A})$. The authors establish a partial converse and, for $p=1$, introduce the cancelling condition on $\mathbb{B}$ restricted to $\ker(\mathscr{A})$, linking to ellipticity plus cancelling in the optimal limiting case. A schematic proof builds on Fonseca–Müller, splitting $P$ into kernel and orthogonal components, combined with constant-rank and injectivity arguments, and culminates in domain-related considerations via $\mathbb{C}$-ellipticity. Overall, the paper provides a unified framework to obtain coercive estimates for a broad class of differential operators, including curl/divergence-type structures, with clear implications for variational methods and boundary-domain problems.

Abstract

Korn-Maxwell-Sobolev (KMS) inequalities represent a tool for estimating differential expressions and have gained particular importance in recent years, especially concerning elliptic operators. In my Master's thesis, together with Peter Lewintan (University of Duisburg-Essen), we extended this concept to also apply to operators of constant rank. This makes it possible to cover more complex structures such as the curl or divergence of vector fields. A key difference from the elliptic theory is that in the constant rank case, a correction term $Π_\mathbb{B}$ is necessary on the left-hand side of the inequality. Results were also obtained for the limiting case $p=1$, although additional assumptions are required here. This article provides an illustrative introduction to KMS inequalities and demonstrates their application in both the elliptic and constant rank cases.