New regularity and uniqueness results in the multidimensional Calculus of Variations

TL;DR

This paper develops new regularity results for polyconvex functionals in elasticity and establishes criteria for the uniqueness of global minimizers, with applications including counterexamples to regularity.

Contribution

It introduces novel regularity theory for polyconvex functionals and provides new uniqueness criteria in finite elasticity, advancing understanding in elasticity calculus of variations.

Findings

Regularity theory for polyconvex functionals in elasticity

Uniqueness criteria for global minimizers in finite elasticity

Counterexample to regularity in elasticity problems

Abstract

In the first part of this doctoral thesis we develop a regularity theory for a polyconvex functional in compressible elasticity. In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity. Here it is our main objective to establish uniqueness criteria, which when present, guarantee the uniqueness of the corresponding global minimizer. Then various applications and generalisations are discussed one of which is the construction of a counterexample to regularity.