2006 Fiscal Year Final Research Report Summary
Research on global properties of solutions of geometric variational problems
| Project/Area Number |
16540195
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | Nara Women's University |
|---|
Principal Investigator |
KOISO Miyuki Nara Women's University, Faculty of Science, Professor, 理学部, 教授 (10178189)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
AIYAMA Reiko (AKUTAGAWA Reiko) University of Tsukuba, Graduate School of Pure and Applied Sciences, Lecturer, 大学院・数理物質科学研究科, 講師 (20222466)
FUJIOKA Atsushi Hitotsubashi University, Graduate School of Economy, Associate Professor, 大学院・経済学研究科, 助教授 (30293335)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | variational problem / anisotropic surface energy / constant anisotropic mean curvature / elliptic parametric functional / Wulff shape / Delaunay surface / capillary surface / rolling construction |
|---|
| Research Abstract |
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.
|
