A Constant Rank Theorem For The Level Set Of A Class Of Mean Curvature Equations

The convexity of the solutions is an important topic in the study of partial differential equations and geometric analysis,which is studied mainly by the macroscopic method and microscopic method. So we naturally want to discuss the convexity of the solutions of the general elliptic and parabolic equations,such as the convexity of solutionss and the convexity of the level sets.The key of the microscopic convexity method in the partial differential equations is to establish the constant rank theorem.In this dissertation,we concentrate on the microscopic convexity of the general elliptic equations,establish the constant rank theorems. The main resuits of this dissertations are as follows.Theorem. Let Ω be a smooth bounded connected domain on the space form Mn with constant curvature ∈≥0.Let u∈C4(Ω) ∩C2((?)) be the solution to the prescribed mean curvature equation where H(x,u) ≥0 satisfies the structure condition Assume |▽u|≠0 in Ω and the level sets of u are all convex with respect to the normal▽u,then the second fundamental form of the level sets of u must have the same rank at all points in Ω.