Calculus of Variations and Geometric Measure Theory

A. Braides - G. Dal Maso

Validity and failure of the integral representation of Γ-limits of convex non-local functionals

created by braidesa on 08 May 2023
modified on 30 Jun 2024

[BibTeX]

Published Paper

Inserted: 8 may 2023
Last Updated: 30 jun 2024

Journal: J. Functional Anal.
Volume: 286
Pages: 110317
Year: 2024
Doi: 10.1016/j.jfa.2024.110317

Abstract:

We prove an integral-representation result for limits of non-local quadratic forms on $H^1_0(\Omega)$, with $\Omega$ a bounded open subset of $\mathbb R^d$, extending the representation on $C^\infty_c(\Omega)$ given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a counterexample showing that a corresponding representation may not hold if we consider analogous functionals in $W^{1,p}_0(\Omega)$, with $p\neq 2$ and $1<p\le d$.

Keywords: Integral representation, Dirichlet forms, non-local functionals


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