Abstract or Additional Information
Optimal partition problems has generated an increasing interest in the past few years in both the fields of applied math and pure math. A partition of a set U in this context is simply a family of N disjoint sets whose union is the whole set U (sometimes they are called N-clusters in literature). An optimal partition problem is a variational problem where we try to minimize a given suitable energy, defined on the partition through a set function on the power set of R^n, eventually under some volume constraint. The most studied problems of these kind are the isoperimetric problem for partitions (where the total energy is given by the perimeter) and the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. The first part of the talk will be focused on a global overview on optimal partition problems and on some interesting issues that arise in the general framework. The second part, instead, will be devoted to the isoperimetric problem for partitions and to the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian.