本文選題：曲率估計(jì)　切入點(diǎn)：水平集　出處：《曲阜師范大學(xué)》2015年碩士論文

【摘要】：橢圓偏微分方程方程解的幾何性質(zhì)的研究是一個(gè)重要主題,特別是解的水平集凸性的曲率估計(jì)是近年來人們非常感興趣的一個(gè)方向.本文介紹了一個(gè)特殊的平均曲率型方程在二維情況下,它的解的水平集凸性的曲率估計(jì).本文主要用了極值原理得到了定理的證明.定理1.1假設(shè)Ω是R2內(nèi)的有界光滑區(qū)域,設(shè)u是平均曲率型方程在Ω中的一個(gè)解,并且u∈C4(Ω)nC2(Ω).如果在Ω上,有|%絬|≠0,且水平集沿外法向%絬是嚴(yán)格凸的,則函數(shù)|%絬|-2K的極小值在邊界上取到,其中K是u的水平集曲率.本文還給出了Heisenberg群中的調(diào)和函數(shù)在次線性增長(zhǎng)條件下Liouville定理的證明.定理1.2設(shè)函數(shù)u是Heisenberg群Hn上的調(diào)和函數(shù),并且滿足次線性增長(zhǎng)則u恒為常數(shù).其中r(ζ)為點(diǎn)ζ到原點(diǎn)的距離.
[Abstract]:The study of geometric properties of solutions of elliptic partial differential equations is an important subject. In particular, the curvature estimation of the level set convexity of solutions is a direction of great interest in recent years. In this paper, we introduce a special mean curvature equation in two dimensions. In this paper, the theorem is proved by the extreme principle. Theorem 1.1 assumes that 惟 is a bounded smooth region in R2, and let u be a solution of the equation of mean curvature type in 惟. And u 鈭，

本文編號(hào)：1662974