The Gagliardo-Nirenberg-Sobolev (GNS) inequalities have played a major role in applied mathematics and mathematical physics in more than half a century. In this talk I will present several new results concerning the counterparts of the GNS inequalities on bounded domains. Of course concentration of the minimizers of the GNS inequalities is a main tool in the proof of existence of minimizers on bounded domains. Naturally concentration occurs in the interior for Dirichlet boundary conditions and on the boundary for Neumann boundary conditions. In the Neumann case this leaves a set of interesting open problems depending on the characteristic of the boundary.
This is in part joint work with Cristobal Vallejos (PUC) and Hanne Van Den Bosch (U. de Chile) and in part with Soledad Benguria (U. Wisconsin, Madison).