Characterization of ellipsoids through an overdetermined boundary value problem of Monge-Ampère type

Barbara Brandolini, Nunzia Gavitone, Carlo Nitsch, Cristina Trombetti

Published 2013-05-10, updated 2013-10-11Version 2

The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.