PDE Seminar Abstracts

Isoperimetric estimates for a class of Dirichlet problems for the Monge-Ampère operator

Carlo Nitsch
University of Naples
3 Aug 2009 3-4pm, Carslaw Room 454


We show that as well as the classical isoperimetric inequalities, affine isoperimetric inequalities can be used to obtain a priori estimates for solutions to elliptic problems. In particular we show that the eigenvalue of the Dirichlet problem for the Monge-Ampère operator, when computed on convex domains with fixed measure, is maximal on ellipsoids. This result is established by exploiting the affine invariant structure of such operator using either Blaschke-Santalò or Petty inequalities.