New isoperimetric estimates for solutions to Monge–Ampère equations

Abstract

We prove some sharp estimates for solutions to Dirichlet problems relative to Monge–Ampère equations. Among them we show that the eigenvalue of the Dirichlet problem, when computed on convex domains with fixed measure, is maximal on ellipsoids. This result falls in the class of affine isoperimetric inequalities and shows that the eigenvalue of the Monge–Ampère operator behaves just the contrary of the first eigenvalue of the Laplace operator.