A characterization of the disc through a hessian equality

TL;DR

This paper characterizes the conditions under which a continuous, boundary-vanishing, thrice differentiable function on a plane domain attains equality in a Hessian inequality, revealing that the domain must be a disc and the function radial.

Contribution

It provides a complete characterization of the equality case in a Hessian inequality, linking it to the domain being a disc and the function being radial.

Findings

Equality in the Hessian inequality holds only for disc-shaped domains.

The function achieving equality must be radial.

The domain's shape is uniquely determined as a disc when equality is attained.

Abstract

Let M be a bounded open plane domain. Let f be a continuous function on the closure of M, 3-times continuously differentiable in M, which vanish on the boundary. Polterovich and Sodin proved that the values of f cannot exceed the norm of the hessian of f, averaged over the entire domain M. In this paper we study the equality case for this inequality. We show that equality holds if and only if M is a open disc and f belongs to a special class of radial functions.