On Polyconvexity and Almgren Uniform Ellipticity With Respect to Polyhedral Test Pairs

TL;DR

This paper investigates the relationship between ellipticity and polyconvexity in anisotropic geometric energy functionals, establishing that uniform ellipticity with respect to polyhedral chains implies polyconvexity of the integrand.

Contribution

It proves that uniform ellipticity of anisotropic energies with respect to polyhedral chains implies polyconvexity of the integrand, extending recent results in the field.

Findings

Uniform ellipticity implies polyconvexity of the integrand.

Extension of recent results by De Rosa, Lei, and Young.

Provides conditions linking ellipticity and polyconvexity in geometric energies.

Abstract

We study anisotropic geometric energy functionals defined on a class of k-dimensional surfaces in a Euclidean space. The classical notion of ellipticity, coming from Almgren, for such functionals is investigated. We prove a variant of a recent result of De Rosa, Lei, and Young and show that uniform ellipticity of an anisotropic energy functional with respect to real polyhedral chains implies uniform polyconvexity of the integrand.