A Class Of Elliptic Equation Solution Level Set Convexity Of The Curvature Estimates

This thesis consists of three sections.The first section is the introduction, and we give the main theorem:LetΩbe a smooth bounded domain in R2 and u∈C4(Ω)∩C2(Q) be a positive solution of the elliptic equation inΩ, i.e.Δu=u+u-1|▽u|2the level sets of u are strictly convex with respect to normal▽u. Then the function u-2|▽u|2k attains its minimum on the boundary. Where k is the curvature of the level sets of u.In the second section, we divide it into two parts. Firstly, we introduce the content and proof about maximum principle. Later, we give the convexity of the graph in differential geom-etry; then introduce brief definitions on the convexity of the level sets and obtain the curvature matrix of the level sets of a function.The third section is the proof of the main theorem. We apply the idea of the priori estimates, by some formal computations for the derivative terms. With the maximum principle, we attain the proof of the main theorem at last.