Calculus of Variations and Geometric Measure Theory
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R. Alicandro - N. Ansini - A. Braides - A. Piatnitski - A. Tribuzio

A variational theory of convolution-type functionals

created by braidesa on 09 Jul 2020

[BibTeX]

Preprint

Inserted: 9 jul 2020
Last Updated: 9 jul 2020

Year: 2020

ArXiv: 2007.03993 PDF

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.


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