**SMS scnews item created by Daniel Daners at Thu 22 Aug 2019 1424**

Type: Seminar

Distribution: World

Expiry: 26 Aug 2019

**Calendar1: 26 Aug 2019 1200-1300**

**CalLoc1: AGR Carslaw 829**

CalTitle1: PDE Seminar: The Brunn-Minkowski inequality for the Monge-Ampere eigenvalue and smoothness of the eigenfunctions (Le)

Auth: daners@dora.maths.usyd.edu.au

### PDE Seminar

# The Brunn-Minkowski inequality for the Monge-Ampere eigenvalue and smoothness of the eigenfunctions

### Le

Nam Quang Le

Indiana University, USA

Mon 26th Aug 2019, 12-1pm, Carslaw Room 829 (AGR)

## Abstract

The original form of the Brunn-Minkowski inequality involves volumes of convex bodies
in ${\mathbb{R}}^{n}$ 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-Ampère operator. For this purpose, I
will first introduce the Monge-Ampère eigenvalue problem on general
bounded convex domains. Then, I will present several properties of the
eigenvalues and related analysis concerning smoothness of the eigenfunctions.

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