Linear nonlocal diffusion problems in metric measure spaces

TL;DR

This paper provides a comprehensive analysis of linear nonlocal diffusion problems across various metric measure spaces, exploring properties like regularity, spectrum, and asymptotic behavior, and comparing them to classical heat equations.

Contribution

It extends the theory of nonlocal diffusion to diverse metric measure spaces, analyzing spectral properties and asymptotic behavior, which was less understood before.

Findings

Established regularity and compactness results for nonlocal operators.

Proved maximum principles for nonlocal diffusion equations.

Described asymptotic behavior using spectral methods.

Abstract

The aim of this paper is to provide a comprehensive study of some linear nonlocal diffusion problems in metric measure spaces. These include, for example, open subsets in $\mathbb{R}^N$, graphs, manifolds, multi-structures or some fractal sets. For this, we study regularity, compactness, positiveness and the spectrum of the stationary nonlocal operator. Then we study the solutions of linear evolution nonlocal diffusion problems, with emphasis in similarities and differences with the standard heat equation in smooth domains. In particular prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.