Mathematics

In the first part of the course I will review the main metric, embedding and structure theorem about Sobolev spaces (depending on the audience more or less in depth), and study the corresponding weak convergence. I will then study lower-semicontinuity conditions for local functionals (that is, integral functionals depending on the weak gradient), described by convexity conditions (convexity, polyconvexity, quasiconvexity, rank-1-convexity), and apply them to obtain solutions of minimum problems by the Direct Method of the Calculus of Variations. I will discuss relaxation results for problems involving non-lower-semicontinuous functionals and state some open problems.In the second part of the course, I will introduce (Riesz) fractional gradients and the related fractional Sobolev spaces. I will prove structure properties of such spaces and highlight differences and analogies with the usual weak gradient. I will define a weak convergence and study weak lower-semicontinuous properties for integral functionals depending on fractional gradients.In the third part of the course I will consider a finite-difference type approach to Sobolev spaces. To that end I will consider double integrals concentrating on the diagonal and prove asymptotic structure properties and Poincaré inequalities. As a result, I will show that local minimum problems on Sobolev spaces can be approximated by problems on Lebesgue spaces involving double integrals. I will discuss lower-semicontinuity and relaxation problems for double integrals and state some open problems.