Maciej Lesniak | Searcharxiv

arXiv2026

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

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~$Q \in \natp$.

math.AP↗

arXiv2025

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

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.

math.AP↗