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Constraint maps and free boundaries

Alessio Figalli, André Guerra, Sunghan Kim, Henrik Shahgholian

Abstract

In this short expository note, we present a selection of classic and recent ideas in free boundary theory, with a focus on the vectorial case, referred to here as constraint maps. The note includes a brief historical perspective and highlights the latest heuristic-level results.

Constraint maps and free boundaries

Abstract

In this short expository note, we present a selection of classic and recent ideas in free boundary theory, with a focus on the vectorial case, referred to here as constraint maps. The note includes a brief historical perspective and highlights the latest heuristic-level results.

Paper Structure

This paper contains 5 sections, 4 theorems, 47 equations, 10 figures.

Table of Contents

  1. A brief history of free boundaries
  2. The scalar obstacle problem
  3. The vectorial obstacle problem
  4. Discontinuities vs free boundaries
  5. Branch points and singular free boundaries

Key Result

Theorem 1

Let ${\mathbf u}\colon \Omega\to {\mathbb R}^m\setminus O$ be a minimizing constraint map, where we assume that Then the function $x\mapsto \operatorname{dist}({\mathbf u}(x), O)$ is continuous.

Figures (10)

  • Figure 1: The solution to Gergonne's problem in the case where we cut a cylinder, rather than a cube, in half.
  • Figure 2: AI-generated King's hat.
  • Figure 3: A possible solution to the obstacle problem
  • Figure 4: Regular vs singular free boundaries: either the contact set looks like a half-space, or it is thin.
  • Figure 5: Courant's problem: the given surface is a torus and the prescribed Jordan curve is in blue.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4