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

TL;DR

This paper provides geometric bounds on the curvature of level sets of solutions to certain nonlinear elliptic equations, and proves a constant rank theorem for their second fundamental form.

Contribution

It introduces a refined structural condition and establishes curvature estimates and a constant rank theorem for convex level surfaces of solutions.

Findings

Established a geometric lower bound for principal curvature.

Proved a constant rank theorem for the second fundamental form.

Applied to solutions of fully nonlinear elliptic equations in convex domains.

Abstract

We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^2u, Du, u, x)=0$ 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.