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

We prove an integral-representation result for limits of non-local quadratic forms on $H_0^1(\mathrm{\Omega })$, with $\mathrm{\Omega }$ a bounded open subset of ${\mathbb{R}}^d$, extending the representation on $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ 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 $W_0^{1,p}(\mathrm{\Omega })$, with $p\mathrm{\ne }2$ and $1<p\le d$.