A Class Of Elliptic Partial Differential Equations Of Convex Curvature Estimation Of Level Set

This dissertation consists of four sections.The first section is the introduction.It introduces the development of the elliptic partial differential equations,and gives the main theorem1.1and theorem1.2.In the second section,we provide the preliminaries in R2and R3.We introduced corre-sponding the different that some fundamental knowledge of graph and its convexity in differ-ential geometry in R2and R3,then we give brief definition of the convex level sets of a func-tion,and derive the curvature matrix of the level sets.At last,some theorems about maximum principle are listed.The third section is preparation for the proof of curvature estimates of the elliptic partial differential equations in R2. The key idea of completing this section is the maximum principle.Let Ω. be a smooth bounded domain in R2and u∈C4(Ω)∩C2(Ω) be a positive solution of the elliptic equation in Ω,i.e. Assume|V▽u|≠0in Ω, and the level sets of u are strictly convex with respect to normal▽u.Let K be the Gaussian curvature of the level sets of u.Then we have the following fact:the function K attains its minimum on the boundary (?)Ω.In the fourth section,we mainly finish some specific computations for the proof of our main theorem.The mainly technique consist of rearranging,the second and the third derivative terms and condition for ψ,we introduce lemma for proof:Let Ω be a smooth bounded domain in R3and u∈C4(Ω)∩C2(Ω) be a positive solution of the elliptic equation in Ω,i.e. we havef>0,inf∈C2(R×R3).Assume for any.χ∈Ω,we have|▽u|×0in Ω, and the level sets of u are strictly convex with respect to normal▽u.Let K be the Gaussian curvature of the level sets of u.Then we have the following fact:the function K attains its minimum on the boundary (?)Ω.