The Geometric Properties Of Solutions For A Class Of Semilinear Elliptic Equations

The problem of the convexity of level set of elliptic partial differential equations has attracted much interest of many mathematicians. In this thesis, we describe that the curvature of level set whose the solutions of a class of elliptic partial equations by minimum principle. There are three parts in this paper:In the first section we suggest the current research state of the level set of elliptic partial equations; in section 2, we give some definitions and lemmas which will be used in the following proofs; Section 3 devoted to the main ideas and the proofs.The following is the main result.Assume Ω be a bounded smooth domain in R~2, u € C~4(Ω) n C~2(Ω) be a solution of equation △u=|▽u|~α. Suppose also|▽u|≠0 in Q,and the level set of u are convex with respect to the normal ▽u. Let k be the curvature of the level sets of u, then when α=0 or α≥5,(ⅰ)|▽u|~2k attains its minimum on the boundary of the domain;(ⅱ)|▽u|~3k attains its minimum on the boundary of the domain.