The Geometric Properties Of The Set Of Solutions Of The Solution Of △ U = G (u) On The Riemannian Manifold

For the study of differential equation, more and more persons have paid attention to the geometric properties of level sets, such as the convexity of level sets, gaussian curvature and principal curvature estimate problem, etc. On the basis of studies which are made, in this paper, we choose the appropriate coordinate system, and then consider the equation which the curvature of level set satisfied on 2-dimensional Riemannian manifolds. At last, we use the maximal principle and the Constant Rank Theorem to study the geometric property of the solution of the equation.According to the content, this article is divided into the following three section:The first section summarizes the development of history and theoretical basis in this paper.The second section discusses several important essential theorems and lemmas.The third section concludes the results based on the calculation.The main results are as follows:Theorem 1. Set Ω ? 2, suppose ∈ 4(Ω) satisfiesSuppose =- log , If() ≥ 0, > 0,then =() is constant rank in Ω and 1 is the first eigenvalue.Theorem 2. Suppose 2is a two-dimensional Riemannian manifold and Ω ? 2, let be a smooth solution of the following semilinear elliptic equation.Suppose satisifies > 0 and ′< 0, If |?| ?= 0 Ω and the level sets of are convex with repect to the normal direction |?|, then the function = |?|3· attains it’s minimum on the boundary.