Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators

TL;DR

This paper develops a theoretical framework for nonlocal boundary value problems governed by symmetric nonlocal operators with general kernels, extending classical approaches to broader classes of nonlocal equations.

Contribution

Introduces a general theory for nonlocal boundary value problems with symmetric kernels, broadening the scope beyond traditional kernel functions.

Findings

Established a Hilbert space approach for weak solutions.

Analyzed the discrete Poisson problem on a unit cube.

Extended classical boundary problem theory to general symmetric kernels.

Abstract

Nonlocal boundary value problems with Dirichlet or Neumann boundary are well-studied for nonlocal operators of the type $\mathcal{L}_\gamma u = \operatorname{PV} \int_{\mathbb{R}^d} \big(u(\cdot)-u(y)\big) \gamma(\cdot,y) \, \mathrm{d}y$ where the underlying kernel function $\gamma: \mathbb{R}^d \times \mathbb{R}^d \rightarrow [0,\infty)$ is assumed to be measurable and symmetric. In this paper, a theory is introduced for problems whose governing operator is of the more general type \[\mathcal{L}u:= \operatorname{PV} \int_{\mathbb{R}^d}\big(u(\cdot)-u(y)\big) \, K(\cdot, \mathrm{d}y)\] where ${K: \mathbb{R}^d \times \mathcal{B}(\mathbb{R}^d) \rightarrow [0,\infty]}$ is a symmetric transition kernel. Our main focus is on nonlocal Dirichlet and Neumann problems and a classical Hilbert space approach is developed for solving designated weak formulations. As an example, the discrete Poisson problem on $\Omega=(0,1)^d$ is discussed.