Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities

TL;DR

This paper establishes sharp conditions for Korn-Maxwell-Sobolev inequalities involving differential operators, extending known estimates to the critical case where the integrability exponent p equals 1, thus resolving an open problem.

Contribution

It provides necessary and sufficient conditions for these inequalities to hold in the borderline case p=1, generalizing previous results and addressing an open problem.

Findings

Sharp conditions for inequalities with p=1 established

Extension of estimates to borderline case p=1 confirmed

Open problem in the field resolved affirmatively

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 M\"{u}ller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative.