Level Set Curvature Estimation For A Class Of Elliptic Equations

This thesis consists of four sections.The first section is the introduction. We introduce the result of the convexity of the level sets of the solutions of elliptic partial differential equations and give the main theorem of the thesis.The second section is the preliminaries. We introduce the curvature of curve and sur-face. It includes the calculating of curvature,the first and the second fundamental form's for surface.Then,we give the way to determine the surface to be convex, and give the definitions of the level sets and the convexity of level sets.Finally,we introduce the curvature matrix of level sets. Some lemmas and propositions are given which are used in the following proof.The third and the fourth section include the proof of the main theorem. We chose suitable function(?),modify the terms of▽(?) with locally bounded coefficients to prove△(?)≤0 mod▽(?) inΩ, by the standard minimum principle, we get the result immediately. The main thought of the the-sis is using the the strong minimum principle to obtain Gaussian curvature estimates for the convex level sets of solutions for elliptic partial differential equations in dimension 2 and 3.