Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

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 $\mathcal{M}^+$.