Inserted: 21 sep 2016
Last Updated: 16 jan 2017
Journal: Journal of Mathematical Analysis and Applications
This paper, accepted in a special volume on convex analysis (proceedings of a conference in Erice, 2016), presents an interesting application of convex duality to regularity, that I also presented in some talks during fall 2015. Beware almost no result is new, only the point of view is new. The same techniques can be applied in a more difficult setting to mean-field games and incompressible fluid mechanics, obtaining non-trivial results.
A technique based on duality to obtain $H^1$ 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.