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Improvement of flatness for nonlocal free boundary problems

Xavier Ros-Oton, Marvin Weidner

TL;DR

This work proves sharp regularity results for the free boundary in the nonlocal one-phase problem driven by general symmetric $2s$-stable operators. By formulating a purely nonlocal viscosity framework and establishing an $\varepsilon$-regularity (improvement-of-flatness) scheme, the authors show that flat free boundaries yield $C^{1,\alpha}$ regularity and that the regular points form an open dense set; in two dimensions, all blow-ups are half-space solutions, yielding global $C^{1,\alpha}$ regularity. The analysis hinges on a nonlocal Bernoulli-type first variation, domain-variation techniques, partial boundary Harnack, and a careful treatment of tail terms that capture long-range interactions. The results extend known fractional-Laplacian phenomena to general nonlocal operators and provide a robust nonlocal methodology with potential applications to related free boundary problems and anisotropic integro-differential operators.

Abstract

In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is $C^{1,α}$ near regular points, and that the set of regular free boundary points is open and dense. Moreover, in 2D we classify all blow-up limits and prove that the free boundary is $C^{1,α}$ everywhere. The main technical tool of our proof is an improvement of flatness scheme, which we establish in the general framework of viscosity solutions, and which is of independent interest. All of these results were only known for the fractional Laplacian, and are completely new for general nonlocal operators. In contrast to previous works on the fractional Laplacian, our method of proof is purely nonlocal in nature.

Improvement of flatness for nonlocal free boundary problems

TL;DR

This work proves sharp regularity results for the free boundary in the nonlocal one-phase problem driven by general symmetric -stable operators. By formulating a purely nonlocal viscosity framework and establishing an -regularity (improvement-of-flatness) scheme, the authors show that flat free boundaries yield regularity and that the regular points form an open dense set; in two dimensions, all blow-ups are half-space solutions, yielding global regularity. The analysis hinges on a nonlocal Bernoulli-type first variation, domain-variation techniques, partial boundary Harnack, and a careful treatment of tail terms that capture long-range interactions. The results extend known fractional-Laplacian phenomena to general nonlocal operators and provide a robust nonlocal methodology with potential applications to related free boundary problems and anisotropic integro-differential operators.

Abstract

In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is near regular points, and that the set of regular free boundary points is open and dense. Moreover, in 2D we classify all blow-up limits and prove that the free boundary is everywhere. The main technical tool of our proof is an improvement of flatness scheme, which we establish in the general framework of viscosity solutions, and which is of independent interest. All of these results were only known for the fractional Laplacian, and are completely new for general nonlocal operators. In contrast to previous works on the fractional Laplacian, our method of proof is purely nonlocal in nature.
Paper Structure (24 sections, 36 theorems, 359 equations)

This paper contains 24 sections, 36 theorems, 359 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of proof
  4. Acknowledgments
  5. Organization of the paper
  6. Preliminaries
  7. Function spaces and solution concepts
  8. Properties of minimizers
  9. Properties of blow-ups
  10. Viscosity solutions to the one-phase problem
  11. The free boundary condition
  12. Viscosity solutions
  13. Optimal regularity for viscosity solutions
  14. Flatness implies $C^{1,\alpha}$ for viscosity solutions
  15. Improvement of flatness
  16. ...and 9 more sections

Key Result

theorem 1

Let $K \in C^{1-2s+\beta}(\mathbb{S}^{n-1})$ for some $\beta > \max\{0,2s-1\}$ and assume eq:Kcomp. Let $u$ be a minimizer of $\mathcal{I}_{\Omega}$ with $B_2 \subset \Omega$. Then, there exists $\delta \in (0,1)$, depending only on $n,s,K$, such that if $0 \in \partial \{ u > 0 \}$ and for some $\n then, $\partial \{ u > 0\} \in C^{1,\alpha}$ in $B_{\rho}$ for any $\alpha \in (0,\frac{s}{2})$, an

Theorems & Definitions (84)

  • theorem 1
  • remark 2
  • theorem 3
  • theorem 4
  • theorem 5
  • remark 6
  • remark 7
  • definition 8: minimizers
  • lemma 9: see Lemma 4.1 in RoWe24b
  • lemma 10: optimal regularity
  • ...and 74 more