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

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~$Q \in \natp$.