Variational problems depending on non-local, non-singular operators. Application to integral equations

We explore variational problems depending on some regular classes of integral operators. We are particularly interested in understanding the connection between the nature of the integral operator and the convexity properties required on the integrand leading to weak lower semicontinuity and existence results, as well as optimality conditions. The application of some of these results to particular integrands permits some existence results for integral equations, in addition to numerical approximation methods based on this variational strategy. We illustrate the benefits of our approach by looking at some selected examples.