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Optimal regularity for nonlocal elliptic equations and free boundary problems

Xavier Ros-Oton, Marvin Weidner

Abstract

In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.

Optimal regularity for nonlocal elliptic equations and free boundary problems

Abstract

In this article we establish for the first time the boundary regularity of solutions to nonlocal elliptic equations with kernels . This was known to hold only when is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal regularity of solutions to obstacle problems for general nonlocal operators with kernels . Again, this was only known when is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal regularity and non-degeneracy estimates.
Paper Structure (24 sections, 41 theorems, 344 equations)

This paper contains 24 sections, 41 theorems, 344 equations.

Table of Contents

  1. Introduction
  2. Obstacle problems for nonlocal operators
  3. 1D solutions
  4. One-phase nonlocal free boundary problems
  5. Strategy of the proof of $C^s$ regularity in bounded domains
  6. Acknowledgements
  7. Organization of the paper
  8. Preliminaries
  9. Function spaces and solution concepts
  10. Harmonic replacement
  11. Homogeneous barriers and $C^{2s-1+\varepsilon}$ estimates
  12. One-phase nonlocal free boundary problem
  13. Existence and basic properties of minimizers
  14. Optimal regularity
  15. Non-degeneracy
  16. ...and 9 more sections

Key Result

theorem 1

Let $\Omega\subset \mathbb{R}^n$ be a $C^{1,\alpha}$ domain, and $L$, $K$, $s$, $\lambda$, and $\Lambda$ be as in eq:L-eq:Kcomp. Let $f\in L^\infty(B_1 \cap \Omega)$, and $u$ be any weak solution of Then, we have where $C$ depends only on $\Omega$, $s$, $\lambda$, $\Lambda$. Moreover, if $u\geq0$ in $\mathbb{R}^n$ and $f\geq0$, then either $u\equiv0$ or with $c>0$, and where $d(x):={\rm dist}(x

Theorems & Definitions (97)

  • theorem 1
  • remark 2
  • theorem 3
  • theorem 4
  • theorem 5
  • theorem 6
  • definition 7: minimizers
  • remark 8
  • definition 9: weak solutions
  • remark 10
  • ...and 87 more