On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations

2013/02/22 by Guy Barles, Barles, Guy, Christine Georgelin +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.1302.5568

openalex publication_date 2013/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Inspired by the penalization of the domain approach of Lions & Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give apropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems.

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• The inhomogeneous fractional Dirichlet problem

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