Positive Solutions Of Elliptic Equations Level Set Convexity Estimates

This dissertation consists of four sections.The first section is the introduction.In the second section, We firstly introduce some fundamental knowledge of graph and its convexity in differential geometry, then we give brief definition of the convex level sets of a function, 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 our main theorem. Depending on refer-ence of related papers, we mainly finish some specific computations for the proof of our main theorem. The main technique consists of rearranging the second and third derivative terms and the first derivative condition forφ. Later, we introduce two lemmas.In the fourth section, the proof of our main theorem is given. The key idea of completing this paper is the application of the priori estimates. We prove the following theorem:LetΩbe a smooth bounded domain in R3 and u∈C4(Ω)∩C2(Ω) be a positive solution of the elliptic equation inΩ, i.e.Δu= f(u,▽u)= u+u-1|▽u|2. Assume|▽u|≠0 inΩ, 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 u-2|▽u|2K attains its minimum on the boundary (?)Ω.