Continuity of some non-local functionals with respect to a convergence of the underlying measures

TL;DR

This paper investigates the stability of certain non-local functionals on Sobolev spaces under measure convergence, establishing a framework for their $ ext{Gamma}$-convergence.

Contribution

It introduces a new measure convergence concept on product spaces that ensures the $ ext{Gamma}$-convergence and stability of non-local Sobolev functionals.

Findings

Established a measure convergence notion implying functional stability.

Proved $ ext{Gamma}$-convergence of non-local functionals under this measure convergence.

Provided a framework for analyzing stability of non-local functionals in Sobolev spaces.

Abstract

We study some non-local functionals on the Sobolev space $W^{1,p}_0(\Omega)$ involving a double integral on $\Omega\times\Omega$ with respect to a measure $\mu$. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of $\Gamma$-convergence of the corresponding functionals.