I will present the recent tools I have developed to prove existence and regularity properties of the critical points of anisotropic functionals. In particular, I will provide the anisotropic extension of Allard's celebrated rectifiability theorem and its applications to the anisotropic Plateau problem. Three corollaries are the solutions to the formulations of the Plateau problem introduced by Reifenberg, by Harrison-Pugh and by Almgren-David. Furthermore, I will present the anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, I will focus on the anisotropic isoperimetric problem: I will provide the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points among finite perimeter sets. Moreover I will derive stability inequalities associated to this rigidity theorem.
Some of the presented results are joint works with De Lellis, De Philippis, Ghiraldin, Gioffré, Kolasinski and Santilli.
In this talk, I present two PDE models of a chemical reaction, and I show that they are two different sides of the same coin: namely, the solutions of one PDE converge to the solutions of the other. The proof of this fact is surprisingly elementary (but not easy), because it just requires some integration by parts. This is based on joint work with Lawrence C. Evans. No chemistry background required.
We will show optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity. This is joint work with Jingang Xiong.
We study the long-time asymptotic behavior of the linearized Euler and nonlinear Navier-Stokes equations close to Couette flow. As a main result we show that suitable forcing breaks asymptotic stability results at the level of the vorticity, but that solutions never the less exhibit convergence of the velocity field. Thus, here linear inviscid damping persists despite instability of the vorticity equations.
In his seminal work from 1979,
Joseph J. Kohn invented
his theory of multiplier ideal sheaves
connecting a priori estimates for the d-bar problem
with local boundary invariants
constructed in purely algebraic way.
I will explain the origin and motivation of the problem,
and how Kohn's algorithm reduces it
to a problem in local geometry
of the boundary of a domain.
I then present my recent work with Sung Yeon Kim
based on the technique of jet vanishing orders,
and show how it can be used to
control the effectivity of multipliers in Kohn's algorithm,
subsequently leading to precise a priori estimates.
Abreu type equations are fully nonlinear, fourth order, geometric PDEs which can be viewed as systems of two second order PDEs, one is a Monge-Ampere equation and the other one is a linearized Monge-Ampere equation. They first arise in the constant scalar curvature problem in differential geometry and in the affine maximal surface equation in affine geometry. This talk discusses the solvability and convergence properties of second boundary value problems of singular, fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the Rochet-Choné model of monopolist's problem in economics. This talk explains how minimizers of the 2D Rochet-Choné model can be approximated by solutions of singular Abreu equations.
Mean field game theory is the study of the limit of Nash
equilibria of differential games when the number of players tends to infinity. It
was introduced by J.-M. Lasry and P.-L. Lions, and independently by P.
Caines, M. Huang and R. Malhame. A fundamental object in the theory is the
master equation, which fully characterizes the limit equilibrium. In this
talk, we will introduce Mean field game and master equations on graphs. We will
construct solutions to both equations and link them to the solution to
a Hamilton-Jacobi equation on graph.