Sobolev regularity for solutions of the Monge-Amp\`ere equation and application to the Semi-Geostrophic system

Sobolev regularity for solutions of the Monge-Amp\`ere equation and application to the Semi-Geostrophic system

Thu, 01/12/2011
12:30 		Guido De Philippis (Scuola Normale Superiore di Pisa) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Sobolev regularity for solutions of the Monge-Ampère equation and application to the Semi-Geostrophic system
I will talk about $ W^{2,1} $ regularity for strictly convex Aleksandrov solutions to the Monge Ampère equation
\[ \det D^2 u =f \]
where $ f $ satisfies $ \log f\in L^{\infty} $. Under the previous assumptions in the 90's Caffarelli was able to prove that $ u \in C^{1,\alpha} $ and that $ u\in W^{2,p} $ if $ |f-1|\leq \varepsilon(p) $. His results however left open the question of Sobolev regularity of $ u $ in the general case in which $ f $ is just bounded away from $ 0 $ and infinity. In a joint work with Alessio Figalli we finally show that actually $ |D^2u| \log^k |D^2 u| \in L^1 $ for every positive $ k $.
If time will permit I will also discuss some question related to the $ W^{2,1} $ stability of solutions of Monge-Ampère equation and optimal transport maps and some applications of the regularity to the study of the semi-geostrophic system, a simple model of large scale atmosphere/ocean flows (joint works with Luigi Ambrosio, Maria Colombo and Alessio Figalli).