The Level Of A Class Of Elliptic Equations Set Of Convex Curvature Estimation

This thesis consists of four sections. The first section is the introduction.we mainly in-troduced conclusions about estimates of convex level sets with respect to elliptic equations’s solution.It dated back to studies of minimal annulus whose boundary consists of two closed convex curves,then the convex level sets of Green function was studyed.Later on,this conclu-sion developed from R3to p-harmonic function in higher dimensions.Related equations were extended from linear to semi-linear,new proofs were replaced by old ones,research perspec-tives were also expanded from qualitative to quantitative.Recently, the research of this field was much more developed.The second section are estimates about level curves of a elliptic equation in R2.In this section,we firstly proofed function φ=logK(x) satisfied the following elliptic inequality: coefficients of (?)φ are Locally bounded here.Then,utilizing the minimum principle,we obtain the first important theorems of this paper.In the third section,we expanded our equation into R3.In order to prepare for the proof of our main theorem we listed fundamental contents of local theory related to classical curved surface,symmetric curvature matrix and it’s characteristic value, fundamental form of level sets as well as the outer normal vector of them.Finally,we gave some important lemmas.At last, the proof of our main theorem is given. The key idea of completing this paper is the application of the priori estimates. We’ve proved the following theorem:Let Ω be a smooth bounded domain in R3and u∈C4(Ω)∩C2(Ω) be a solution of the elliptic equation in Ω, i.e. Assume|(?)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 the function K(x) attains its minimum on the boundary (?)Ω.