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.
The issue here is the large time behaviour of the solutions of the Fisher-KPP equation. A well-known result of Kolmogorov-Petrovskii-Piskunov (1937) asserts that a solution whose initial value is the Heaviside function will converge, for large times, to a travelling wave of slowest speed. The profile of the solution is shifted by a strictly sub-linear in time quantity, that is not quantified in KPP's work. Much later, Bramson (1983) proved that the shift was actually logarithmic in time. We propose in this talk to revisit this result, and to discuss how it generalises to equations whose coefficients are inhomogeneous in space. Joint work with F. Hamel, J. Nolen and L. Ryzhik.
Nonlinear elliptic and parabolic PDEs have applications to image processing, first arrival times in wave propagation, homogenization, mathematical finance, stochatic control and games theory. Convergent numerical schemes are important in these applications in order to capture geometric features such as folds and corners, and avoid artificial singularities which arise from bad representations of the operators. In many cases these equations are considered too difficult to solve, which is why linearized models or other approximations are commonly used. Progress has recently been made in building solvers for a class of Geometric PDEs. I'll discuss a few important geometric PDEs which can be solved using a numerical method called Wide Stencil finite difference schemes: Monge-Ampere, Convex Envelope, Infinity Laplace, Mean Curvature, and others. Focusing in on the Monge-Ampere equation, which is the seminal geometric PDE, I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. Several groups of researchers have proposed numerical schemes which fail to converge, or converge only in the case of smooth solutions. I'll present a convergent solver which which is fast: comparable to solving the Laplace equation a few times. The most effective notion of weak solutions for fully nonlinear elliptic equations is that of viscosity solutions, developed by Crandall, Ishii, and Lions. Viscosity solutions enjoy strong stability properties, and allow for uniform convergence of approximations, using the Barles-Souganidis theorem. This theory is used to prove convergence of the finite difference method. The talk will be accessible to graduate students.
