基於羅森－歐斯曼錐構造的均曲流自相似解

Abstract

在這篇論文中，我們首先得到基於羅森－歐斯曼錐構造的均曲流自相似解必須滿足的等式，並證明了自擴張解的存在性。主要的關鍵是利用羅森－歐斯曼錐的對稱性將偏微分方程轉化為常微分方程組，並研究這種近似於自治系統的常微分方程組。特別地，我們發現從狄利克雷問題的觀點來看，我們構造的自擴張解具唯一性。

In this thesis, we derived the equation of self-similar solutions to mean curvature flow based on the Lawson-Osserman cone and proved the existence of self-expander. The main point is to use the symmetry to transform the PDE into a system of ODEs and analyze such analogous autonomous system. In particular, the self-expander is unique form the viewpoint of Dirichlet problem.