Calculus of Variations and Geometric Measure Theory

R. Alicandro - N. Ansini - A. Braides - A. Piatnitski - A. Tribuzio

A Variational Theory of Convolution-type Functionals

created by braidesa on 09 Jul 2020
modified on 12 Oct 2023



Inserted: 9 jul 2020
Last Updated: 12 oct 2023

Journal: SpringerBriefs on PDEs and Data Science
Pages: 121
Year: 2023
Links: book page at


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.