Equivalence of Uniform Polyconvexity and Almgren Uniform Ellipticity for Lipschitz $Q$-Graph Test Pairs

TL;DR

This paper proves the equivalence between uniform polyconvexity and Almgren's uniform ellipticity for Lipschitz $Q$-graph test pairs, advancing the understanding of geometric integrand properties in calculus of variations.

Contribution

It establishes the equivalence between uniform polyconvexity and Almgren's uniform ellipticity for Lipschitz $Q$-graph test pairs, extending previous results and connecting classical and $Q$-integrand concepts.

Findings

Uniform polyconvexity is equivalent to Almgren's uniform ellipticity for Lipschitz $Q$-graph test pairs.

For classical integrands, uniform polyconvexity is equivalent to uniform quasiconvexity of the $Q$-integrand.

The results strengthen the theoretical framework relating geometric integrand properties.

Abstract

We investigate the relationship between uniform polyconvexity of anisotropic geometric integrands and Almgren's uniform ellipticity. We first establish the converse implication for uniform ellipticity with respect to polyhedral test pairs, thereby strengthening earlier results. Our main theorem shows that uniform polyconvexity is equivalent to Almgren's uniform ellipticity with respect to Lipschitz $Q$-graph test pairs, building on techniques developed by De Rosa, Lei, and Young. As a consequence, we show that for a classical integrand, uniform polyconvexity is equivalent to uniform quasiconvexity of the associated $Q$-integrand for every~$Q \in \natp$.