Geometric Properties Of Convexity Set Of Minimal Surface On Two - Dimensional Manifold

Partial differential equations may be supposed to one of the most interesting studies. It is many Math lovers who are absorbed in the geometrical properties of partial differential equations. As is known to us,there are lots of interesting results about the curvature estimates of the level sets of the solutions of elliptic partial differential equations.A lemma related to transformation of higher derivative is given, as a useful tool,we will study the geometrical properties of the level sets of a quasilinear equations. For an auxiliary function on two dimensional Riemannian manifolds involving the curvature of the level sets,we obtain the curvature estimate eventually.Theorem 1. Let Ω be a smooth bounded domain in R2. Let u∈C4(Ω)∩C2(Ω) be the solution of the following minimal surface equation Assume|▽u|≠0 in Ω.If the level sets of u are strictly convex with respect to the normal direction ▽u,and let K be the curvature of the level sets. Then attain its minimum and maximum on the boundary (?)Ω.This theorem is proved in the prime paper written by Wang-Zhang. The following are the results of this paper:Theorem 2. Let Ω be a smooth bounded connected domain on the two dimensional space form M2 with constant curvature ε. Let u∈C4(Ω)∩C2(Ω) be the minimal graph defined on Ω, i.e. div =0. Assume |▽u|≠0 in Ω and the level sets of u are convex with respect to the normal ▽u, then they are strictly convex.Theorem 3. Let Ω be a smooth bounded connected domain on the two dimensional space form M2 with constant curvature ε. Let u∈C4(Ω)∩C2(Ω) such that div K be the curvature of the level sets of u, assume|▽u|≠0 in Ω. Then we have(i) For ε>0, the function K attains its minimum on the boundary (?)Ω;(ii) For ε=0, the function 1/2K attains its maximum and minimum on the boundary (?)Ω;(iii) For ε<0, the function attains its maximum on the boundary (?)Ω.