Korn-Maxwell-Sobolev inequalities for general incompatibilities

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 Müller (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 $\mathrm{inc}$.

arXiv comment33 pages, 3 figures

References in corpus (15)

• A unifying perspective: the relaxed linear micromorphic continuum
• On Rank-One Convex Functions that are homogeneous of Degree One
• Limiting Sobolev inequalities for vector fields and canceling linear differential operators
• An elastoplastic theory of dislocations as a physical field theory with torsion
• Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy
• On the Trace Operator for Functions of Bounded $\mathbb{A}$-Variation
• On limiting trace inequalities for vectorial differential operators
• Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields
• Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
• Hilbert Complexes with Mixed Boundary Conditions -- Part 2: Elasticity Complex
• $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative
• $L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions
• A canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin accounting for the Burgers vector
• Matrix representation of a cross product and related curl-based differential operators in all space dimensions
• Fractional Korn and Hardy-type inequalities for vector fields in half space

Cited by in corpus (2)

• A computational approach to identify the material parameters of the relaxed micromorphic model
• A global higher regularity result for the static relaxed micromorphic model on smooth domains