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

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 $1<p\le d$.