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A remark on staircase laminates in restricted sets

Igor Buchowiec, Pholphum Kamthorntaksina, Katarzyna Mazowiecka, Armin Schikorra, Akshara Vincent

Abstract

We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, Müller, Székelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Székelyhidi on optimal Meyers' regularity theory via this framework.

A remark on staircase laminates in restricted sets

Abstract

We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, Müller, Székelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Székelyhidi on optimal Meyers' regularity theory via this framework.
Paper Structure (5 sections, 10 theorems, 187 equations)

This paper contains 5 sections, 10 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Notion of reducibility and proof of Theorem \ref{['th:thm4.1']}
  3. The staircase laminate criterion --- Proof of Theorem \ref{['th:thm4.3']}
  4. Application to the Astala-Faraco-Székelyhidi result: Proof of Theorem \ref{['th:AFS']}
  5. Proof of Theorem \ref{['th:AFS']}

Key Result

Theorem 1.1

For any $\Lambda > 1$ there exists a bounded measurable symmetric matrix $A\colon \mathbb{B}^2 \to \mathbb{R}^{2 \times 2}$, with a function $u \in W^{1,2}(\mathbb{B}^2, \mathbb{R})$, and an affine linear map $l \colon \mathbb{R}^2\to\mathbb{R}$ such that and for any $p \geq \frac{2\Lambda}{\Lambda -1}$.

Theorems & Definitions (26)

  • Theorem 1.1: AFS08
  • Theorem 1.2: Exact solution, cf. KMSX24
  • Theorem 1.3: Staircase laminate criterion, cf. KMSX24
  • Definition 2.1: Regular domain
  • Definition 2.2: Piecewise affine map
  • Definition 2.3: Reducibility and exact reducibility in weak $L^p$
  • Lemma 2.4
  • proof : Proof of \ref{['la:exactPART']}
  • proof : Proof of \ref{['th:thm4.1']}
  • Definition 3.1: Elementary splitting and elementary splitting with construction steps
  • ...and 16 more