Continuity of some non-local functionals with respect to a convergence of the underlying measures (Q2111987)

In the article some non-local functionals on the Sobolev space \(W^{1,p}_0(\Omega)\) involving a double integral on \(\Omega\times \Omega\) with respect to a measure \(\mu\) are studied. The authors introduce a notion of convergence of measures on product spaces which implies a stability property in the sense of the \(\Gamma\)-convergence of the corresponding functionals with respect to the weak topology in \(W^{1,p}_0(\Omega)\). Under some additional assumptions, the convergence of the same finctionals in the sense of Mosco convergence in \(W^{1,p}_0(\Omega)\) is also obtained.