The Lower Bound Of The Curvature Of The Horizontal Set Of The Extremely Class-like Hypersurface

For a long time, many scholars devote to studying the geometric properties of the solutions to elliptic partial differential equations. In this direction, researching of the level sets with to relevant properties is an important and interesting contents, many meaningful results have been obtained from these work. This paper will give a similar result with [22] about the curvature of level sets of the solution to the space-like maximal surface equation, namely the curvature of the level sets of space-like maximal surface equation with solution attains its minimum on the boundary. The detailed proof will be given in this paper. This paper will be divided into three sections-introduction, preliminaries and the proof of theorem. In the second section, we firstly introduce the origin of the maximal surface equation, that is to say I will give a reduction of the surface equation in the Minkowski's space, the maximum principle which we will use during the proof and the curvature formulas and so on. The third section mainly give the proof of the theorem.Theorem 1.2 Assume that ? is a smooth bounded domain in R~2, and let u €C~4(?)? C~2(?) be ? space-like maximal surface defined in ? such that Moreover, Suppose that|?u|?0 and the level sets of u are convex with respect to Du. Then we obtain that where K is the curvature of the level set for u in ?.