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 H01(Ω), with Ω a bounded open subset of Rd, extending the representation on Cc∞(Ω) 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 W01,p(Ω), with p≠2 and 1<p⩽d.