Sharp mean Hadamard inequalities and polyconvex integrands that give rise to convex functionals

TL;DR

This paper explores mean Hadamard inequalities in two dimensions, establishing the uniqueness of minimizers for certain convex functionals with computational support.

Contribution

It proves the uniqueness of minimizers for integral functionals with polyconvex integrands under mixed boundary conditions, based on new mean Hadamard inequalities.

Findings

Proved mean Hadamard inequalities in two dimensions.

Established uniqueness of minimizers for polyconvex integrals.

Supported results with computational experiments.

Abstract

We investigate several instances of the Hadamard inequality in the mean in two dimensions. As a consequence, we prove the uniqueness of minimizers of an integral functional with a polyconvex integrand, subject to mixed Dirichlet and Neumann boundary conditions. The theoretical findings are complemented by computational experiments that illustrate the behavior of the minimizers.