A Type of Nonlocal Elliptic Problem: Existence and Approximation Through a Galerkin--Fourier Method

Abstract

The aim of this paper is to study a type of nonlocal elliptic equation whose format includes a kernel $k$ and a design function $h$. We analyze how this equation is connected with the classical elliptic equation that includes $h$ as diffusive term. On one hand, the spectrum of the nonlocal operator that defines the nonlocal equation is studied. Existence and unicity of solutions for the nonlocal equation are proved. On the other hand, the convergence of these solutions to the solution of the classical elliptic equation as the kernel $k$ converges to a Dirac Delta is analyzed. This work is performed by using an spectral theorem on the nonlocal operator and by applying some specific compactness results. The kernel $k$ is assumed to be radial. Dirichlet boundary conditions are assumed for the classical problem, whereas for the nonlocal equation a nonlocal boundary Dirichlet constraint must be defined.