Compactness for a class of integral functionals with interacting local and non-local terms

TL;DR

This paper establishes a compactness result for a class of integral functionals combining local and non-local terms, with implications for concentration and homogenization, highlighting the interaction effects in the limit.

Contribution

It introduces a novel compactness theorem for integral functionals with local and non-local interactions, considering their combined effects in the $ ext{Gamma}$-limit.

Findings

Proves a $ ext{Gamma}$-convergence compactness result for mixed local and non-local functionals.

Shows the local part of the limit depends on the interaction between terms.

Applies the result to concentration and homogenization problems.

Abstract

We prove a compactness result with respect to $\Gamma$-convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the $\Gamma$-limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.