Shape optimization problems with Robin boundary conditions.

In this talk I will discuss shape optimization problems with Robin boundary conditions on the free part. For general shape functionals, the existence of a solution may not occur, but for a suitable class of energy type functionals one can prove the existence of a solution and some partial regularity results. The main tools are of free discontinuity type. As example, one could consider the minimization of the first Robin eigenvalue of the Laplacian, among arbitrary open sets of prescribed measure, contained in a given design region.