Korn-Maxwell-Sobolev inequalities for general incompatibilities

TL;DR

This paper develops a broad family of Korn-type inequalities for incompatible fields with superlinear growth, extending previous results to more general incompatibilities and higher-order scenarios, with applications in continuum mechanics.

Contribution

It introduces sharp, generalized Korn inequalities for incompatible fields, including new results involving Kröner's incompatibility tensor and higher-order incompatibilities.

Findings

Established coercive Korn-type inequalities for general incompatible fields.

Extended inequalities to higher-order incompatibilities and more general incompatibility measures.

Provided sharp inequalities applicable to continuum mechanics models with dislocation energies.

Abstract

We establish a family of coercive Korn-type inequalities for generalised incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work (ArXiv 2206.10373), where we focussed on the case $p=1$ and incompatibilities governed by the matrix curl, the case $p>1$ considered in the present paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalisations of recently proved inequalities by the last two authors and M\"{u}ller (Calc. Var. PDE 60 (2021), 150) in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher order scenarios as well, our approach equally gives the first and sharp inequalities involving Kr\"{o}ner's incompability tensor $\mathrm{inc}$.