A variational theory of convolution-type functionals

TL;DR

This paper develops a variational framework for convolution-type functionals, establishing their limits as local integral functionals with p-growth, with applications in homogenization, point-clouds, and gradient flows.

Contribution

It introduces a general variational theory for convolution energies, proving compactness and integral representation results for their limits.

Findings

Limits of convolution-type energies are local integral functionals.

The framework applies to periodic and stochastic homogenization.

Results extend to functionals on point-clouds and gradient flows.

Abstract

We provide a general treatment of perturbations of a class of functionals modeled on convolution energies with integrable kernel which approximate the $p$-th norm of the gradient as the kernel is scaled by letting a small parameter $\varepsilon$ tend to $0$. We first provide the necessary functional-analytic tools to show coerciveness in $L^p$. The main result is a compactness and integral-representation theorem which shows that limits of convolution-type energies is a standard local integral functional with $p$-growth defined on a Sobolev space. This result is applied to obtain periodic homogenization results, to study applications to functionals defined on point-clouds, to stochastic homogenization and to the study of limits of the related gradient flows.