Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations

Abstract

We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^{2}u,Du,u,x)=0$ in convex ring domains, under a refined structural condition introduced by Bianchini-Longinetti-Salani in \cite{BLS}. We also prove a constant rank theorem for the second fundamental form of the convex level surfaces of these solutions.