The seminar meets regularly on wednesday at 4pm in Eckhart 202. We also have special seminars during other days.
The Monge-Ampère equation arises in connections with several problems from geometry and analysis (optimal transport, the Minkowski problem, the affine sphere problem, etc.) The regularity theory for this equation has been widely studied. In particular, in the 90's Caffarelli developed a regularity theory for Alexandrov/viscosity solutions, showing that strictly convex solutions of $det(D^2u)=f$ are $C^{1,\alpha}$ provided $f$ is bounded away from zero and infinity, and are $W^{2,p}$ with $p>1$ if $f$ is uniformly close to a constant. The counterexamples by Wang in 1995 showed that the results of Caffarelli were more or less optimal. However, an important question which remained open was whether solutions with right hand side bounded away from zero and infinity could be $W^{2,1}$. In a recent joint work with De Philippis we proved that this is the case: indeed, not only solutions are $W^{2,1}$, but that actually $|D^2 u|$ belongs to $L\log^{k}L$ for any $k >0$. As an application, this result allows to prove global existence of distributional solutions to the semigeostrophic equations (joint work with Ambrosio, Colombo, and De Philippis). In this talk I'll first briefly describe the connection between Monge-Ampère and the semigeostrophic equations, and then I'll focus on the proof of the $W^{2,1}$ regularity.
In this joint work with G. Carlier and B. Nazaret, we study a family of distances between probability measures recently introduced by Dolbeault, Nazaret and Savare. We show that computing such distances is equivalent to solving mean-field-game-like systems. In one space dimension, we also describe the geometry of the geodesics associated with these distances and show their relation with some degenerate elliptic equations.
In the talk we address a system of PDEs describing an interaction between an incompressible fluid and an elastic body. The fluid motion is modeled by the Navier-Stokes equations while an elastic body evolves according to an linear elasticity equation. On the common boundary, the velocities and stresses are matched. We discuss available results on local well-posedness and prove new existence and uniqueness results with the velocity and the displacement belonging to low regularity spaces.
