A microscopic convexity principle for nonlinear partial differential equations

We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian ∇ ² u is of constant rank for any convex solution u of equation F(∇ ² u,∇ u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed. Research of the first author was supported in part by NSFC No.10671144 and National Basic Research Program of China (2007CB814903). Research of the second author was supported in part by an NSERC Discovery Grant.