Regularity for nonlocal equations with local Neumann boundary conditions

Ros-Oton, Xavier; Weidner, Marvin

doi:10.2140/apde.2026.19.353

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Regularity for nonlocal equations with local Neumann boundary conditions

Xavier Ros-Oton and Marvin Weidner

Vol. 19 (2026), No. 2, 353–411

DOI: 10.2140/apde.2026.19.353

Abstract

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 $v ≍ {dist}^{s - 1}$ and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients $v ∕ {dist}^{s - 1}$ , 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.

Keywords

nonlocal, regularity, boundary, large solution, Neumann condition

Mathematical Subject Classification

Primary: 31B05, 35B44, 35B65, 35R35, 47G20

Milestones

Received: 26 March 2024

Revised: 31 October 2024

Accepted: 2 February 2025

Published: 22 January 2026

Authors

Xavier Ros-Oton
ICREA and Departament de Matemàtiques i Informàtica Universitat de Barcelona Spain

Marvin Weidner
Departament de Matemàtiques i Informàtica Universitat de Barcelona Spain	Institute for Applied Mathematics University of Bonn Germany

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Vol. 19, No. 2, 2026