Wulff shape symmetry of solutions to overdetermined problems for Finsler Monge-Amp\`ere equations

TL;DR

This paper investigates symmetry properties of solutions to anisotropic Monge-Ampère equations, demonstrating that solutions exhibit Wulff shape symmetry under certain overdetermined boundary conditions, extending classical symmetry results to anisotropic settings.

Contribution

It establishes the Wulff shape symmetry of solutions to overdetermined Monge-Ampère type equations modeled on general anisotropic norms, generalizing classical symmetry results.

Findings

Solutions are symmetric with respect to the Wulff shape associated with the norm H.

The analysis extends symmetry results to anisotropic Monge-Ampère equations.

Overdetermined boundary conditions enforce Wulff shape symmetry of solutions.

Abstract

We deal with Monge-Amp\`ere type equations modeled upon general anisotropic norms $H$ in $\mathbb R^n$. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject to both a homogeneous Dirichlet condition and a second boundary condition, designed on $H$, on the gradient image of the domain. The Wulff shape symmetry associated with $H$ of the solutions is established.