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Regularity for nonlocal equations with local Neumann boundary conditions

Xavier Ros-Oton, Marvin Weidner

TL;DR

This work develops a sharp boundary regularity theory for nonlocal equations with local Neumann boundary data, focusing on large boundary blow-up solutions that behave like $v\sim d^{s-1}$. The authors prove a Schauder-type estimate: if $\Omega$ is $C^{k+1,\gamma}$ and the kernel $K$ is sufficiently smooth, then the quotient $v/d^{s-1}$ lies in $C^{k,\gamma}_{\text{loc}}(\overline{\Omega})$, with quantitative bounds that depend on the data and domain regularity, while avoiding the critical case $\gamma=s$. The strategy combines a new weak maximum principle for nonlocal Neumann problems, a boundary Hölder theory via a growth/weak-Harnack framework, and a blow-up argument anchored by a Liouville theorem in the half-space with Neumann data to obtain higher-order boundary regularity; barriers adapted to the $d^{s-1}$ blow-up and the notion of nonlocal equations up to a polynomial are key technical components. The results have direct implications for nonlocal free boundary problems, connecting boundary regularity to the behavior of linearized problems near the free boundary and enabling a localized, scale-aware analysis. Overall, the paper extends Schauder-type boundary regularity to a nonlocal Neumann blow-up setting and provides tools for localized, higher-order regularity and free-boundary applications.

Abstract

In this article we establish fine results on the boundary behavior of solutions to nonlocal equations in $C^{k,γ}$ domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like $v \asymp d^{s-1}$ and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients $v/d^{s-1}$, depending on the regularity of the domain and on the data of the problem. The results of this article will be important in a forthcoming work on nonlocal free boundary problems.

Regularity for nonlocal equations with local Neumann boundary conditions

TL;DR

This work develops a sharp boundary regularity theory for nonlocal equations with local Neumann boundary data, focusing on large boundary blow-up solutions that behave like . The authors prove a Schauder-type estimate: if is and the kernel is sufficiently smooth, then the quotient lies in , with quantitative bounds that depend on the data and domain regularity, while avoiding the critical case . The strategy combines a new weak maximum principle for nonlocal Neumann problems, a boundary Hölder theory via a growth/weak-Harnack framework, and a blow-up argument anchored by a Liouville theorem in the half-space with Neumann data to obtain higher-order boundary regularity; barriers adapted to the blow-up and the notion of nonlocal equations up to a polynomial are key technical components. The results have direct implications for nonlocal free boundary problems, connecting boundary regularity to the behavior of linearized problems near the free boundary and enabling a localized, scale-aware analysis. Overall, the paper extends Schauder-type boundary regularity to a nonlocal Neumann blow-up setting and provides tools for localized, higher-order regularity and free-boundary applications.

Abstract

In this article we establish fine results on the boundary behavior of solutions to nonlocal equations in domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients , depending on the regularity of the domain and on the data of the problem. The results of this article will be important in a forthcoming work on nonlocal free boundary problems.
Paper Structure (18 sections, 29 theorems, 329 equations)

This paper contains 18 sections, 29 theorems, 329 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. A weak maximum principle and nonlocal problems with local Dirichlet conditions
  4. Strategy of the proof: regularity for nonlocal problems with local Neumann data
  5. Applications to free-boundary problems
  6. Acknowledgments
  7. Organization of the paper
  8. Preliminaries
  9. Function spaces and solution concepts
  10. Nonlocal operators and the distance function
  11. Barriers with boundary blow-up
  12. Nonlocal equations up to a polynomial
  13. Two lemmas on viscosity solutions
  14. Nonlocal maximum principles with local Dirichlet and Neumann conditions
  15. Hölder estimates up to the boundary
  16. ...and 3 more sections

Key Result

theorem 2

Let $L$, $K$, $s$, $\lambda$, $\Lambda$ be as in eq:L-eq:K. Let $k \in \mathbb{N}$, $\gamma \in (0,1)$ with $\gamma \not= s$, and $\Omega \subset \mathbb{R}^n$ be a $C^{k+1,\gamma}$ domain, and $K \in C^{2k+2\gamma+3}(\mathbb{S}^{n-1})$. Let $v \in L^1_{2s}(\mathbb{R}^n)$ with $v/d^{s-1} \in C(\over where $\nu : \partial \Omega \to \mathbb{S}^{n-1}$ is the normal vector of $\Omega$, and $f \in C(\

Theorems & Definitions (64)

  • remark 1
  • theorem 2
  • proposition 3
  • theorem 4
  • proposition 5
  • theorem 6
  • theorem 7
  • remark 8
  • definition 9: Viscosity solution
  • lemma 10
  • ...and 54 more