Willmore演進方程之等高集描述

Abstract

吾人考慮Willmore 演進方程之等高集描述。我們曾利用均曲率演進方程之等高集描述的構想 來推導Willmore 演進方程。然而我們發現在弱解的定義上即無法突圍，此問題主要根源於此 方程為四階擬線性方程組。最近 Droske 與Rumpf 在其數值模擬中採取另一種等高集描述方 法模擬出橢圓到球面的演進。本計畫旨在導出兩種Willmore 演進方程之等高集描述。其一為 Droske 與Rumpf 的模式。在適當逼近的考量下，另一為源於保角高斯映射的等高集描述。 期望在這兩種描述中能找出定義弱解的可能，希望藉以能討論弱解的存在性。

The proposal of this project is to find a suitable level set formulation for the Willmore flow in the 3-dimensional Euclidean space. Because of the Willmore flow is a system of fourth order quasilinear evolution equations, we found that the level set formulation for the Willmore flow analogous to the weak formulation for the mean curvature flow constructed by Evans and Spruck does not work, we cannot obtain the existence of the viscosity solution. Recently, Droske and Rumpf give a new approach in level set form. In accordance with numerical simulation, they observe that an initial function with ellipse-like level sets evolved under Willmore flow tend to get rounder. Here we want to consider two different types of formulations. The first one is that of Droske and Rumpf, we want to find a formulation in differential form. For finding an approximation appropriate for the weak solution, the second one is the envelope of a family of oriented contact spheres that rises in the conformal Gauss map. The existence of viscosity solution shall come from a well-posed formulation.