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Two-phase almost minimizers for a fractional free boundary problem

Mark Allen, Mariana Smit Vega Garcia

Abstract

In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries $F^+(u)=\partial\{u(\cdot,0)>0\}$ and $F^-(u)=\partial\{u(\cdot,0)<0\}$ cannot touch, that is, $F^+(u)\cap F^-(u)=\emptyset$. Lastly, we prove a flatness implies $C^{1,γ}$ result for the free boundary.

Two-phase almost minimizers for a fractional free boundary problem

Abstract

In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries and cannot touch, that is, . Lastly, we prove a flatness implies result for the free boundary.
Paper Structure (15 sections, 30 theorems, 240 equations)

This paper contains 15 sections, 30 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. The Alt-Caffarelli-Friedman Functional
  3. The Fractional Laplacian and Caffarelli-Silvestre Extension
  4. Almost Minimizers
  5. Main Results
  6. Outline of the Paper
  7. Acknowledgments
  8. Optimal regularity
  9. Nondegeneracy
  10. Separation of the free boundaries
  11. Comparison subsolutions and supersolutions
  12. Compactness
  13. Free Boundary Regularity
  14. Future directions
  15. Minimizers are viscosity solutions

Key Result

Theorem 1

Let $u$ be an almost minimizer of $J$ on $B_1$ with $0<s<1$. Then $u\in C^{0,s}(B_1)$.

Theorems & Definitions (59)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem : Optimal Regularity, see Theorem \ref{['T:s3']}
  • Theorem : Separation of free boundaries, see Theorem \ref{['t:sep1']}
  • Theorem : Flatness implies $C^{1,\gamma}$, see Theorem \ref{['t:reg']}
  • Proposition 2.1
  • Lemma 2.2: Lemma 3.1 from JP2
  • Theorem 2.3
  • ...and 49 more