Characterization of ellipsoids through an overdetermined boundary value problem of Monge-Amp\`ere type

TL;DR

This paper proves that solutions to a specific overdetermined boundary value problem of Monge-Ampère type are ellipsoids, using maximum principles and affine curvature flow techniques.

Contribution

It establishes that all solutions to this nonlinear boundary problem must be ellipsoids, revealing a new symmetry result for Monge-Ampère equations.

Findings

Solutions are ellipsoids due to symmetry properties.

Maximum principle and entropy estimates are key tools.

Results connect boundary value problems with geometric shapes.

Abstract

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.