Regularity via duality in calculus of variations and degenerate elliptic PDEs

A technique based on duality to obtain H1 or other Sobolev regularity results for solutions of convex variational problems is presented. This technique, first developed in order to study the regularity of the pressure in the variational formulation of the Incompressible Euler equation, has been recently re-employed in Mean Field Games. Here, it is shown how to apply it to classical problems in relation with degenerate elliptic PDEs of p-Laplace type. This allows to recover many classical results via a different point of view, and to have inspiration for new ones. The applications include, among others, variational models for traffic congestion and more general minimization problems under divergence constraints, but the most interesting results are obtained in dynamical problems such as Mean Field Games with density constraints or density penalizations.