Wulff shape symmetry of solutions to overdetermined problems for Finsler Monge-Ampère equations

Abstract

We deal with Monge-Ampère 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.