A quantitative constant rank theorem for quasiconcave solutions to fully nonlinear elliptic equations

Abstract

We investigate the fully nonlinear elliptic equations F(D2u,Du,u,x)=0, which satisfy the structural condition previously posed by Bianchini-Longinetti-Salani in 2009. By establishing a novel differential inequality, we prove a weak Harnack inequality for the principal curvatures of the level surfaces of the solutions. This result is indeed a quantitative version of the constant rank theorem showed by Guan-Xu in 2013.