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.

我们在尖锐标准下为超线性增长体系中的广义不相容域建立了一系列强制科恩型不等式。这以最佳方式扩展并统一了先前已知的几个不等式，这些不等式对于连续介质力学中多种模型的存在理论至关重要。与我们之前的工作不同[F. Gmeineder、P. Lewintan 和 P. Neff，所有维度上的最优不相容 Korn-Maxwell-Sobolev 不等式，计算。变种。 PDE  62 (2023) 182]，我们重点关注案例$p=1$和由矩阵旋度控制的不兼容性，情况$p>1$本文中考虑的问题使我们能够从谐波分析中获得更强的结果，但相反地处理更普遍的不兼容性。特别是，我们获得了最近由 P. Lewintan、S. Müller 和 P. Neff 证明的不等式的尖锐概括[具有共形不变位错能的三个空间维度中不相容张量场的科恩不等式，计算。变种。 PDE  60 (2021) 150] 在具有共形不变位错能的不相容科恩型不等式领域。然而，由于也适用于高阶场景，我们的方法同样给出了涉及 Kröner 不相容张量inc的第一个和尖锐的不等式。