Research on global properties of solutions of geometric variational problems

几何变分问题解的全局性质研究

基本信息

批准号：
16540195
负责人：
KOISO Miyuki
金额：
$ 1.86万
依托单位：
Nara Women's University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

1. We studied critical points of an anisotropic surface energy for immersed surfaces in the euclidean three-space with a volume constraint. We assume that the surface energy satisfies a certain "convexity condition". In this case, the energy is called a constant coefficient elliptic parametric functional, and the critical points are surfaces with constant anisotropic mean curvature (CAMC surfaces). The energy-minimizer is a smooth convex surface which is called the Wulff shape (up to translation and homothety). In the case where the anisotropic surface energy is rotationally invariant, we obtained the following results.(1) We proved that any embedded closed CAMC surface was (up to translation and homothety) the Wulff shape.(2) We studied capillary surfaces for certain rotationally invariant elliptic parametric functionals supported on two horizontal planes separated by a fixed distance. When the wetting energy is nonnegative, we proved the existence and uniqueness of the stable solution. Also, we determined the geometric property of the solution. When the wetting energy is negative, we obtained some criterions for the existence and uniqueness of the stable solution. Also, we obtained some numerical results on this problem.2. We obtained a new method of constructing examples of CAMC surfaces whose surface energy is not necessarily rotationally invariant. It will be useful not only for the study on CAMC surfaces but also for other fields such as crystallography, mathematical biology, and so on.3. Anisotropic Delaunay surfaces are surfaces of revolution with constant anisotropic mean curvature. e showed how the generating curves of such surfaces could be obtained as the trace of a point attached to a curve which was rolled without slipping along a line. This generalizes the classical construction for constant mean curvature surfaces due to Delaunay. Moreover, we characterize anisotropic Delaunay surfaces by using their isothermic self-duality.

1。我们研究了与体积约束的欧几里得三空间中浸泡表面的各向异性表面能的临界点。我们假设表面能满足一定的“凸条件”。在这种情况下，能量称为恒定系数的椭圆形参数函数，临界点是具有恒定各向异性平均曲率（CAMC表面）的表面。能量最小化器是一种平滑的凸表面，称为wulff形状（直至翻译和同感）。在各向异性表面能在旋转不变的情况下，我们获得了以下结果。（1）我们证明，任何嵌入的封闭的CAMC表面都是（符合翻译和同种）WULFF形状。（2）我们研究了毛细血管表面，以通过在两个地平面上支撑一个旋转椭圆型函数，以分为两个地平面的距离。当润湿能量不负时，我们证明了稳定溶液的存在和独特性。另外，我们确定了溶液的几何特性。当润湿能为负时，我们获得了一些稳定溶液的存在和独特性的标准。另外，我们在此问题上获得了一些数值结果。2。我们获得了一种构建CAMC表面示例的新方法，CAMC表面不一定旋转不变。它不仅对于对CAMC表面的研究，而且对其他领域（例如晶体学，数学生物学等）也有用。3。各向异性Delaunay表面是持续各向异性平均曲率的革命表面。 e显示了如何获得此类表面的生成曲线，作为连接到曲线的点的痕迹，该曲线是在不沿线滑动的情况下滚动的。这概括了由于Delaunay引起的恒定平均曲率表面的经典结构。此外，我们通过使用等温线性自由度来表征各向异性的Delaunay表面。