2003 Fiscal Year Final Research Report Summary
Functinal analytic approach to reseaches on deformation and stability of surfaces with constant mean curvature
| Project/Area Number |
13640211
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Nara Women's University (2003)
Kyoto University of Education (2001-2002) |
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Principal Investigator |
KOISO Miyuki Nara Women's University, Faculty of Science, Professor, 理学部, 教授 (10178189)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIOKA Atsushi Hitotsubashi University, Graduate School of Economics, Associate Professor, 大学院・経済学研究科, 助教授 (30293335)
AIYAMA Reiko (相山 玲子) University of Tsukuba, Institute of Mathematic, Lecturer, 数学系, 講師 (20222466)
MIYAOKA Reiko Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (70108182)
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| Project Period (FY) |
2001 – 2003
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| Keywords | constant mean curvature / prescribed mean curvature / variational problem / Gauss map / stability / parametric elliptic functional |
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| Research Abstract |
1.We studied the variational theory of surfaces whose mean curvature is prescribed to be a linear function of their height above a horizontal plane (PMC surfaces). We developed a flux formula and used it to prove nonexistence results for closed PMC surfaces. The perturbation theory for PMC surfaces was studied. We obtained necessary conditions for the stability of PMC surfaces with planar boundaries.
2.We posed a variational problem for surfaces whose solutions are a geometric model for thin films with gravity which is partially supported by a given contour. The energy functional contains surface tension, a gravitational energy and a wetting energy, and the Euler-Lagrange equation can be expressed in terms of the mean curvature of the surface, the curvatures of the free boundary and a few other geometric quantities. Especially, we studied in detail a simple case where the solutions were vertical planar surfaces bounded by two vertical lines. We determined the stability or instability of each solution.
3.We studied the geometry of surfaces which are in equilibrium for an anisotropic surface energy with a volume constraint. We obtained the first and second variations and studied the exceptional set of the Gauss map for such surfaces. Also we obtained representation formulas of the equilibrium surfaces of revolution and studied the geometry and stability of these surfaces.
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