Title: The Brunn-Minkowski inequality for the Monge-Ampere eigenvalue and smoothness of the eigenfunctions
Speaker: Nam Le - Indiana University
Abstract: The original form of the Brunn-Minkowski inequality involves volumes of convex bodies in R^n and states that the n-th root of the volume is a concave function with respect to the Minkowski addition of convex bodies. In 1976, Brascamp and Lieb proved a Brunn-Minkowski inequality for the first eigenvalue of the Laplacian. In this talk, I will discuss a nonlinear analogue of the above result, that is, the Brunn-Minkowski inequality for the eigenvalue of the Monge-Ampere operator. For this purpose, I will first introduce the Monge-Ampere eigenvalue problem on general bounded convex domains. Then, I will present several properties of the eigenvalues and related analysis concerning smoothness of the eigenfunctions.