Existence and non-existence of solutions for a class of Monge-Ampère equations

Zhitao Zhang
Academy of Mathematics and Systems Sciences, The Chinese Academy of Sciences, Beijing
12 Oct 2009 3-4pm, Carslaw Room 454

Abstract

We study the boundary value problems for a class of Monge-Ampère equations: det D²u = e^-u in Ω ⊂ Rⁿ,n ≥ 1, u| _∂Ω = 0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we reduce the equation to an ODE, and show there exists a critical radius such that if the radius of a ball is smaller than this critical value, then there exists a solution and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. We calculate the one dimensional case explicitly, which also indicates some kind of bifurcation phenomena may exist. Finally for the fixed domain we consider an equivalent problem with a parameter det D²u = e^-tu in Ω, u| _∂Ω = 0,t ≥ 0. By using Lyapunov-Schmidt Reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimates and bifurcation theory we get the global structure and prove existence of at least two solutions for a certain range of parameters t (0,T^*), a unique solution at t = T^*, nonexistence of solution for t > T^*.

Check also the PDE Seminar page. Enquiries to Florica Cîrstea or Daniel Daners.