Korn's second inequality and geometric rigidity with mixed growth conditions

TL;DR

This paper extends geometric rigidity and Korn's inequality to mixed growth spaces, crucial for nonlinear elasticity and plasticity, by establishing new inequalities in $L^p+L^q$ and $L^{p,q}$ spaces.

Contribution

It introduces generalized Korn's inequality and geometric rigidity results in mixed growth spaces, broadening their applicability in nonlinear elasticity and plasticity theories.

Findings

Proved geometric rigidity in $L^p+L^q$ and $L^{p,q}$ spaces.

Generalized Korn's inequality to these mixed growth spaces.

Established foundational inequalities for applications in nonlinear elasticity.

Abstract

Geometric rigidity states that a gradient field which is $L^p$-close to the set of proper rotations is necessarily $L^p$-close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in $L^p+L^q$ and in $L^{p,q}$ interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn's inequality to these spaces.