Minimizers for nonlocal energy functionals

The emergence of nonlocal theories as promising models in different areas of science (continuum mechanics, biology, image processing) has led the mathematical community to conduct varied investigations of systems of integro-differential equations. In this talk I will present some recent results on regularity of solutions to integral equations with weakly singular kernels. This work is part  of a developing theory that is the nonlocal counterpart of elliptic regularity; indeed, many nonlocal results mimic well-known properties and theorems that hold for elliptic systems. The implications could have far-reaching applications in mathematics, at both theoretical and applied levels.