$\mathcal{A}$-quasiconvexity, G\r{a}rding inequalities and applications in PDE constrained problems in dynamics and statics

TL;DR

This paper establishes a Gårding inequality for $\\mathcal{A}$-quasiconvex functions, extending weak-strong uniqueness and minimality results in PDE constrained problems in dynamics and statics, with applications to the curl operator case.

Contribution

It introduces a new Gårding inequality for $\\mathcal{A}$-quasiconvex functions and applies it to extend uniqueness and minimality results in PDE constrained problems.

Findings

Proved a Gårding inequality for $\\mathcal{A}$-quasiconvex quadratic forms.

Extended weak-strong uniqueness results to PDE constrained problems.

Proved uniqueness of $L^p$ local minimisers in the curl case.

Abstract

A G\r{a}rding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimisers. Using this new G\r{a}rding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$ local minimisers in the classical $\mathcal{A}= {\rm curl}$ case.