Mathematics

The course is subdivided in three parts, which may vary in size and content according to the interests of the audience. 1. Nonlocal discrete systems. We consider simple boundary value problems on sets of integers depending on a function minimizing a nonconvex interaction potential and a long-range convex energy. Our goal is to describe the interaction between nonconvexity and nonlocality through asymptotic properties of solutions and of minima as the size of the domain of minimization diverges. Some lessons will be devoted to analogous problems from the standpoint of dynamical systems;2. Convergence of gradient flows for free-discontinuity problems.We consider minimizing-movements in one dimension for the Mumford-Shah functional and some of its approximations, showing convergence and non-convergence results;3. Relaxation of nonlocal energies.We analyze the lower semicontinuous envelope of non-convex energies depending on a double integrals, which leads to non-integral functionals.