一类椭圆方程正解的水平集凸性的曲率估计

【作者】 李春香；

【导师】 王培合；

【作者基本信息】 曲阜师范大学 ， 基础数学， 2011， 硕士

【摘要】 本文共分三节.第一节为本文的引言,同时给出了主要定理：设Ω是R2中的一有界光滑区域,u∈C4(Ω)∩C2(Ω),且u是椭圆方程Δu=u+u-1|▽u|2在Ω中的一个正解,若u的水平集对于外法向量▽u是严格凸的,则函数u-2|▽u|2k在边界可以取到极小值.其中,K是u的水平集的曲率.第二节中的预备知识我们分成了两部分.第一部分介绍了极大值原理的内容以及证明；接下来的第二部分简单叙述了微分几何学中图的凸性定义和函数水平集凸性的概念,给出了函数水平集的曲率矩阵.第三节为主要定理的证明,运用先验估计的思想,通过一系列的求导计算分析,根据第二节中的极大值原理使定理得以证明.更多还原

【Abstract】 This thesis consists of three sections.The first section is the introduction, and we give the main theorem:LetΩbe a smooth bounded domain in R2 and u∈C4(Ω)∩C2(Q) be a positive solution of the elliptic equation inΩ, i.e.Δu=u+u-1|▽u|2the level sets of u are strictly convex with respect to normal▽u. Then the function u-2|▽u|2k attains its minimum on the boundary. Where k is the curvature of the level sets of u.In the second section, we divide it into two parts. Firstly, we introduce the content and proof about maximum principle. Later, we give the convexity of the graph in differential geom-etry; then introduce brief definitions on the convexity of the level sets and obtain the curvature matrix of the level sets of a function.The third section is the proof of the main theorem. We apply the idea of the priori estimates, by some formal computations for the derivative terms. With the maximum principle, we attain the proof of the main theorem at last.更多还原