Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

TL;DR

This paper investigates symmetry properties of viscosity solutions to fully nonlinear elliptic equations, linking spectral characteristics of an associated operator to the symmetry of solutions, with implications for eigenfunction properties.

Contribution

It introduces a novel connection between the spectral properties of a linearized operator and the symmetry of viscosity solutions for fully nonlinear equations.

Findings

Viscosity solutions exhibit foliated Schwarz symmetry under certain spectral conditions.

The spectral bounds of the associated operator provide insights into the structure of eigenfunctions.

The results extend symmetry analysis to nonlinear elliptic equations with rotational invariance.

Abstract

We study symmetry properties of viscosity solutions of fully nonlinear uniformly elliptic equations. We show that if $u$ is a viscosity solution of a rotationally invariant equation of the form $F(x,D^2u)+f(x,u)=0$, then the operator $\mathcal{L}_u=\mathcal{M}^++\frac{\partial f}{\partial u}(x,u)$, where $\mathcal{M}^+$ is the Pucci's sup--operator, plays the role of the linearized operator at $u$. In particular, we prove that if $u$ is a solution in a radial bounded domain, if $f$ is convex in $u$ and if the principal eigenvalue of $\mathcal{L}_u$ (associated with positive eigenfunctions) in any half domain is nonnegative, then $u$ is foliated Schwarz symmetric. We apply our symmetry results to obtain bounds on the spectrum and to deduce properties of possible nodal eigenfunctions for the operator $\mathcal{M}^+$.