The nonlocal Neumann problem

2022/08/09 by Leonhard Frerick, Frerick, Leonhard, Christian Vollmann +3 · 1 citation
Computer Science · Mathematics · #35A23 #46E35 #47G10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2208.04561

openalex publication_date 2022/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The classical local Neumann problem is well studied and solutions of this problem lie, in general, in a Sobolev space. In this work, we focus on nonlocal Neumann problems with measurable, nonnegative kernels, whose solutions require less regularity assumptions. For kernels of this kind we formulate and study the weak formulation of the nonlocal Neumann problem and we investigate a nonlocal counterpart of the Sobolev space H¹ as well as a resulting nonlocal trace space. We further establish, mainly for symmetric kernels, various existence results for the weak solution of the Neumann problem and we discuss related necessary conditions. Both, homogeneous and nonhomogeneous Neumann boundary conditions are considered. In addition to that, we present a new weak formulation of a Robin problem, where we reformulate the Robin problem into a regional problem.

Cited by

• Asymptotic Compatibility of the Approximate-Ball Finite Element Method for 2D Nonlocal Poisson Problem with Neumann Boundary Conditions

Related

• Nonlocal problems with Neumann and Robin boundary condition in fractional Musielak-Sobolev spaces
• Robust nonlocal trace spaces and Neumann problems
• A remark on nonlocal Neumann conditions for the fractional Laplacian
• Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces
• On Neumann Type Problems for nonlocal Equations set in a half Space