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Non-local diffusion with free boundaries

Diego Marcon, Rafayel Teymurazyan

TL;DR

The paper addresses a non-local free boundary problem driven by the fractional diffusion operator $(-\Delta)^s$ and targets optimal regularity and geometric properties of minimizers. It develops a three-parameter penalization approach to handle obstacle and volume constraints, proves existence and uniform estimates, and passes to the limit to obtain an $s$-Hölder continuous solution with optimal regularity. The authors then show that for small penalization, the solution solves the original problem and inherits regularity results, including non-degeneracy, finite exterior boundary measure, and local $C^{1,\alpha}$ regularity for interior and exterior free boundaries. Collectively, the results extend non-local free boundary theory to kernels comparable to the fractional Laplacian, providing sharp geometric and regularity conclusions without relying on the extension argument.

Abstract

We prove optimal regularity and derive several geometric properties for solutions of a free boundary problem with fractional diffusion. Additionally, we deduce local $C^{1,α}$ regularity results for the corresponding interior and exterior free boundaries.

Non-local diffusion with free boundaries

TL;DR

The paper addresses a non-local free boundary problem driven by the fractional diffusion operator and targets optimal regularity and geometric properties of minimizers. It develops a three-parameter penalization approach to handle obstacle and volume constraints, proves existence and uniform estimates, and passes to the limit to obtain an -Hölder continuous solution with optimal regularity. The authors then show that for small penalization, the solution solves the original problem and inherits regularity results, including non-degeneracy, finite exterior boundary measure, and local regularity for interior and exterior free boundaries. Collectively, the results extend non-local free boundary theory to kernels comparable to the fractional Laplacian, providing sharp geometric and regularity conclusions without relying on the extension argument.

Abstract

We prove optimal regularity and derive several geometric properties for solutions of a free boundary problem with fractional diffusion. Additionally, we deduce local regularity results for the corresponding interior and exterior free boundaries.

Paper Structure

This paper contains 8 sections, 19 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and first results
  3. Notations
  4. L-infinity-bounds
  5. Uniform estimates
  6. Hölder regularity of minimizers
  7. Back to the original problem
  8. Regularity of the free boundaries

Key Result

Proposition 2.1

The functional $I_{\sigma,\delta,\varepsilon}:H^s(\mathbb{R}^n) \longrightarrow \mathbb{R}$, defined by 2.1, has a minimizer.

Theorems & Definitions (38)

  • Proposition 2.1
  • proof
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.2
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • ...and 28 more