Korn–Maxwell–Sobolev inequalities for general incompatibilities

We establish a family of coercive Korn-type inequalities for generalized 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 [F. Gmeineder, P. Lewintan and P. Neff, Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions, Calc. Var. PDE 62 (2023) 182], where we focused on the case $p=1$ and incompatibilities governed by the matrix curl, the case $p>1$ considered in this paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalizations of recently proved inequalities by P. Lewintan, S. Müller and P. Neff [Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy, 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 Kröner’s incompability tensor inc.