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

Maciej Lesniak

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~$Q \in \natp$.

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 -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 -integrand for every~.
Paper Structure (10 sections, 14 theorems, 115 equations)

This paper contains 10 sections, 14 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric Integrands
  4. Classical Integrands
  5. Lipschitz $Q$-valued functions
  6. $Q$-graphs of Lipschitz functions
  7. Ellipticity
  8. Main Result
  9. $Q$-Energies
  10. Uniform Quasiconvexity

Key Result

Theorem 1

Let $\Psi : \mathbf{G}_0(n,k) \to \mathbf{R}$ be a Lipschitz geometric integrand and let $c>0$. Then $\Psi$ is Almgren uniformly elliptic with constant $c$ with respect to polyhedral $k$-chains with real coefficients if and only if $\Psi$ is uniformly polyconvex with the same constant $c$.

Theorems & Definitions (75)

  • Theorem : see \ref{['thm:main1']}
  • Theorem : see \ref{['corollary:thm2']}
  • Theorem : see \ref{['thm:main3']} for the precise statement
  • Definition 2.1
  • Remark 2.2: Federer1969
  • Definition 2.3: Federer1969
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6: Federer1969
  • Definition 2.7: cf. Rockafellar1970
  • ...and 65 more