Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularity and compactness for critical points of degenerate polyconvex energies

André Guerra, Riccardo Tione

Abstract

We study Lipschitz critical points of the energy $\int_Ωg(\det D u) \, d x$ in two dimensions, where $g$ is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers in particular an open problem posed by Kirchheim, Müller and Šverák in 2003.

Regularity and compactness for critical points of degenerate polyconvex energies

Abstract

We study Lipschitz critical points of the energy in two dimensions, where is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers in particular an open problem posed by Kirchheim, Müller and Šverák in 2003.
Paper Structure (6 sections, 10 theorems, 63 equations)

This paper contains 6 sections, 10 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Notation and setup
  3. Exact solutions
  4. Approximate solutions
  5. Differential inclusions and Young measures
  6. Stability and compactness of approximate solutions

Key Result

Theorem 1.1

Suppose that $g\in C^1(\mathbb{R})$ is strictly convex and let $u\in W^{1,\infty}(\Omega,\mathbb{R}^2)$ be a solution of eq:EL. Then $\det \textup{D} u$ is constant a.e. in $\Omega.$

Theorems & Definitions (11)

  • Theorem 1.1: Regularity of exact solutions
  • Corollary 1.2: Critical points are minima
  • Theorem 1.3: Compactness of approximate solutions
  • Corollary 1.4: Quasiconvexity of the differential inclusion
  • Theorem 3.1: Renormalized solutions
  • Definition 4.1
  • Theorem 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 4.5
  • ...and 1 more