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Global regularity and free boundary geometry in the planar Choné-Rochet model

Shibing Chen, Alessio Figalli, Yi Ru-Ya Zhang

Abstract

In this paper, we study minimizers of the Choné--Rochet variational problem in dimension two. We first establish global $C^1$ regularity on arbitrary bounded convex domains, and then prove global $C^{1,1}$ regularity on bounded strictly convex domains or, more generally, whenever the zero set of $u$ has positive measure. Next, we construct smooth bounded convex domains with a flat boundary segment for which no prescribed modulus of continuity controls the gradient; this shows that, without additional geometric assumptions, global $C^1$ regularity is optimal. Finally, we prove that the tamed free boundary (that is, the interface between the strictly convex and non-strictly convex regions of the solution) is locally a $C^1$ embedded curve, significantly strengthening previously known regularity results.

Global regularity and free boundary geometry in the planar Choné-Rochet model

Abstract

In this paper, we study minimizers of the Choné--Rochet variational problem in dimension two. We first establish global regularity on arbitrary bounded convex domains, and then prove global regularity on bounded strictly convex domains or, more generally, whenever the zero set of has positive measure. Next, we construct smooth bounded convex domains with a flat boundary segment for which no prescribed modulus of continuity controls the gradient; this shows that, without additional geometric assumptions, global regularity is optimal. Finally, we prove that the tamed free boundary (that is, the interface between the strictly convex and non-strictly convex regions of the solution) is locally a embedded curve, significantly strengthening previously known regularity results.
Paper Structure (18 sections, 21 theorems, 374 equations, 1 figure)

This paper contains 18 sections, 21 theorems, 374 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation of the model
  3. A related work
  4. Structure of the paper
  5. Global regularity estimates
  6. Preliminary regularity away from two-ended leaves
  7. Global $C^1$ regularity
  8. Global $C^{1,1}$ regularity in strictly convex domains
  9. A counterexample to the global $C^{1,1}$ regularity
  10. Construction of the domain
  11. Zeros on the left flat side
  12. A one-dimensional region near the left flat side
  13. One-dimensional reduction and blow-up of $v"$ when $\delta=0$
  14. Positivity of the slope at the left boundary for large $a$
  15. Existence of a blow-up configuration
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb R^2$ be a bounded convex domain. Then

Figures (1)

  • Figure 1: Schematic of the convex domain $\Omega_{\epsilon,a}=\{(x_1,x_2):a\le x_1\le a+1,\ |x_2|\le \epsilon+g(x_1-a)\}$. The curved parts represent the graphs $x_2=\pm(\epsilon+g(x_1-a))$ with vertical tangents at $x_1=a$ and $x_1=a+1$. The thick segments indicate the flat portions of the boundary.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • ...and 32 more