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 11 Jan 2024


Published Paper (on line)

Inserted: 8 may 2023
Last Updated: 11 jan 2024

Journal: J. Functional Anal.
Year: 2024
Doi: 10.1016/j.jfa.2024.110317


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