A Variational Theory of Convolution-Type Functionals

We provide a general treatment of a class of functionals modeled on convolution energies with kernel having finite p-moments. Such model energies approximate the p-th norm of the gradient as the kernel is scaled by letting a small parameter epsilon tend to 0. We first provide the necessary functional-analytic tools to show coerciveness of families of such functionals with respect to strong Lp convergence. The main result is a compactness and integral-representation theorem which shows that limits of convolution-type energies are local integral functionals with p-growth defined on a Sobolev space. This result is applied to obtain periodic homogenization results, to study applications to functionals defined on point-clouds, to stochastic homogenization and to the study of limits of the related gradient flows.