Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

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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,D²u)+f(x,u)=0, then the operator Lᵤ=M++∂ f/∂ u(x,u), where 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 Lᵤ (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 M+.