Level Set Convexity Of Elliptic Partial Differential Equations Of A Priori Estimates

The thesis consists of four sections.The first section is the introduction. It mainly introduces some achievements and develop-ment about convexity of level sets of elliptic partial diferential equation. At the same time, weintroduce what we do in the thesis,that is we discuss its curvature estimates for the level sets ofa special elliptic partial diferential equation in R2and R3.The second section are preliminaries. It focus on the maximum principle and some relatedproofs.Furthermore,we introduce some theory such as the definition of the convex level sets andthe curvature matrix of the level sets of a function.In the third and the fourth section, we make the priori estimate for the level sets curvature ofthe solution of a special elliptic partial diferential equation. In order to obtain the conclusion,weuse a subsidiary function, maximum principle, and geometry on boundary. It is shown thatLet be a smooth bounded domain in R2or R3and u∈C4() C2(ˉ) be a solution ofthe elliptic equation in, i.e.Assume|u|0in, and the level sets of u are strictly convex with respect to normal u. LetK be the curvature or Gaussian curvature of the level sets of u.Then we have the following fact:the function K attains its minimum on the boundary.