Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators

Abstract

Nonlocal boundary value problems with Dirichlet or Neumann boundary are well-studied for nonlocal operators of the type ${\mathcal{L}}_{\hat{I}}^{3}u=\mathrm{PV}{\int }_{{\mathbb{R}}^d}(u(\cdot )-u(y)){\hat{I}}^3(\cdot ,y)\mathrm{d}y$ where the underlying kernel function ${\hat{I}}^3:{\mathbb{R}}^d\times {\mathbb{R}}^d\to [0,\infty )$ 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 $K:{\mathbb{R}}^d\times \mathcal{B}({\mathbb{R}}^d)\to [0,\infty ]$ 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 $\hat{I}\text{c◯}=(0,1{)}^d$ is discussed.