Séance Séminaire

Séminaire des Doctorant·e·s

MPxMP : When the Moving Plane meets the Maximum Principle

In this talk, I will talk about PDEs while trying to be understood by non-specialists. Rather than presenting some recent results, I will focus on a classical and beautiful theorem. The proof rests on a technic, called the "Moving Plane Method", introduced by Alexandroff in the 40s for geometrical matters, combined with a PDE tool called the « Maximum Principle ». The method is very elegant, robust, and give a grasp of the deep links between geometry and PDEs. In a first step, I will briefly introduce the Maximum Principle, try to give a grasp of it, and infer a couple of corollaries which will be useful in the sequel. The Maximum Principle is the real core of the theory of elliptic PDEs. Then, I will give a proof of the Gidas-Ni-Niremberg Theorem (1979): Let B be the unit ball in R^n, f a Lipschitz function and u a C^2 function, positive solution of -?u=f(u) in B and u=0 on the border of B. Then u is radial and du/dr<0. This is a fundamental result, and in my opinion the proof is one of the most beautiful we can find in the elliptic PDE theory.