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We establish a family of coercive Korn-type inequalities for generalisedincompatible fields in the superlinear growth regime under sharp criteria. Thisextends and unifies several previously known inequalities that are pivotal tothe existence theory for a multitude of models in continuum mechanics in anoptimal way. Different from our preceding work (ArXiv 2206.10373), where wefocussed 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 substantiallystronger results from harmonic analysis but conversely deals with more generalincompatibilities. Especially, we obtain sharp generalisations of recentlyproved inequalities by the last two authors and M\"{u}ller (Calc. Var. PDE 60(2021), 150) in the realm of incompatible Korn-type inequalities withconformally invariant dislocation energy. However, being applicable to higherorder scenarios as well, our approach equally gives the first and sharpinequalities involving Kr\"{o}ner's incompability tensor $\mathrm{inc}$.

Korn-Maxwell-Sobolev inequalities for general incompatibilities