調和映射熱流之度等式與固有值不等式

Abstract

本論文分為兩部份: 第一部份證明二個二維球間之調和映射的熱流解在流經爆炸點時，解的映射度將被提昇 或削減，且其被提昇或削減的量為所有爆炸調和球的映射度之總和。因此，此熱流弱解 最終所收斂之調和映射的映射度，可由起始映射的映射度與熱流過程中所有的爆炸調和 球得出。 第二部份旨在探討$n$維球面上具特定凸性條件的平滑區域之笛里西與諾伊曼固有值之間 的不等式。此部份證明:若邊界具非負均曲率，則對於任意$k=1,2,\cdots$， 第$k$個非零諾伊曼固有值將小於等於第$k$個笛里西固有值。並且如果更進一步 加強條件使邊界具非負第二基本型，則對於任意$k=1,2,\cdots$， 第$k+\left[\frac{n-1}{2}\right]$個非零 諾伊曼固有值將小於等於第$k$個笛里西固有值。

This thesis is divided into two parts. In the first part, we prove that the degree of the solution to the heat equation for harmonic maps between 2-spheres will be increasing or decreasing by the sum of the degrees of the harmonic spheres through each blow-up time. Thus the degree of the harmonic limit will be precisely determined from the degree of the initial map and the amount of the degrees of finite harmonic spheres. The purpose of the second part is to describe some inequalities between Dirichlet and Neumann eigenvalues for smooth domains in the $n$-sphere under certain convex restrictions on the boundary. We prove that if the mean curvature of the boundary is nonpositive, then the $k$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k = 1, 2, \cdots $. Furthermore, if the second fundamental form of the boundary is nonpositive, then the $(k+\left[\frac{n-1}{2}\right])$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k = 1, 2, \cdots $.