Radial solutions for equations of Weingarten type

TL;DR

This paper investigates radial solutions to a class of fully nonlinear Weingarten equations, establishing existence, non-existence, explicit solutions, and symmetry results depending on the elliptic, hyperbolic, or parabolic nature of the PDE.

Contribution

It provides new existence, non-existence, explicit solutions, and symmetry results for radial solutions of Weingarten equations in different PDE regimes.

Findings

Existence of radial solutions in elliptic case for small disks

No radial solutions in hyperbolic case

Explicit solutions in parabolic case

Abstract

In this paper we study the linear Weingarten equation defined by the fully non-linear PDE $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\, \frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=\phi\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$ in a domain $\Omega\subset\mathbb{R}^2$, where $\phi\in C^1([-1,1])$ and $a,b\in\mathbb{R}$. We approach the existence of radial solutions when $\Omega$ is a disk of small radius, giving an affirmative answer when the PDE is of elliptic type. In the hyperbolic case we show that no radial solution exists, while in the parabolic case we find explicitly all the solutions. Finally, in the elliptic case we prove uniqueness and symmetry results concerning the Dirichlet problem of such equation.