A general framework for nonlocal Neumann problems

Within the framework of Hilbert spaces, we solve nonlocal problems in bounded domains with prescribed conditions on the complement of the domain. Our main focus is on the inhomogeneous Neumann problem in a rather general setting. We also study the transition from exterior value problems to local boundary value problems. Several results are new even for the fractional Laplace operator. The setting also covers relevant models in the framework of peridynamics.

Keywords

Neumann problem, nonlocal Sobolev spaces, integro-differential operators, integrodifferential equations, Dirichlet forms

2010 Mathematics Subject Classification

28A80, 35J20, 35J92, 46B10, 46E35, 47A07, 49J40, 49J45

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Financial support for Guy Foghem by the DFG via IRTG 2235, “Searching for the regular in the irregular: Analysis of singular and random systems” is gratefully acknowledged. Guy Foghem also gratefully acknowledges financial support by the DFG via the Research Group 3013: “Vector-and Tensor-Valued Surface PDEs”.

Financial support for Moritz Kassmann by the DFG via IRTG 2235, “Searching for the regular in the irregular: Analysis of singular and random systems” is gratefully acknowledged.

Received 25 May 2022

Accepted 5 May 2023

Published 7 December 2023