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Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

Bruce Kleiner, Stefan Müller, László Székelyhidi, Xiangdong Xie

Abstract

We develop a general toolbox to study $W^{1,p}$ solutions of differential inclusions $\nabla u \in K$ for unbounded sets $K$. A key notion is the concept that a subset $K$ of the space $\mathbb{R}^{d \times m}$ of $d \times m$ matrices can be reduced to another set $K'$. We then use this framework to show that the product rigidity for Sobolev maps fails for $p<2$, and also apply our toolbox to simplify several examples from the literature.

Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

Abstract

We develop a general toolbox to study solutions of differential inclusions for unbounded sets . A key notion is the concept that a subset of the space of matrices can be reduced to another set . We then use this framework to show that the product rigidity for Sobolev maps fails for , and also apply our toolbox to simplify several examples from the literature.
Paper Structure (27 sections, 31 theorems, 201 equations)

This paper contains 27 sections, 31 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Strategy of the proof
  4. Lipschitz differential inclusions
  5. Basic definitions and tools
  6. Laminates
  7. Laminates and differential inclusions
  8. Staircase laminates
  9. Staircase laminates and differential inclusions
  10. Properties of staircase laminates
  11. Examples of staircase laminates
  12. Example 1
  13. Example 2
  14. Example 3
  15. Example 4
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose that $n \ge 2$, $f \in W^{1,2}(\Omega;\mathbb{R}^{2n})$ and that the weak differential $\nabla f(x)$ is split and bijective for a.e. $x \in \Omega$. Then $f$ is split.

Theorems & Definitions (66)

  • Theorem 1.1: otherpaper
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 3.1: Staircase laminates
  • proof
  • ...and 56 more