2012 Fiscal Year Final Research Report
Diffusion and Geometry of Domain
| Project/Area Number |
20340031
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tohoku University (2012)
Hiroshima University (2008-2011) |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
MISAWA Masashi 熊本大学, 大学院・自然科学研究科, 教授 (40242672)
MIKAMI Toshio 広島大学, 大学院・工学研究院, 教授 (70229657)
SHIBATA Tetsutaro 広島大学, 大学院・工学研究院, 教授 (90216010)
KANON Yukio 愛媛大学, 教育学部, 教授 (00177776)
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| Co-Investigator(Renkei-kenkyūsha) |
KAWAKAMI Tatsuki 大阪府立大学, 大学院・工学研究科, 講師 (20546147)
ISHIGE Kazuhiro 東北大学, 大学院・理学研究科, 教授 (90272020)
OGAWA Takayoshi 東北大学, 大学院・理学研究科, 教授 (20224107)
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| Project Period (FY) |
2008 – 2012
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| Keywords | 熱方程式 / 非線形拡散方程式 / 非線形楕円型方程式 / 不変な等位面 / 領域の幾何 / 球面の特徴付け / 超平面の特徴付け / 円柱面の特徴付け |
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| Research Abstract |
Partial differential equations describing diffusion phenomena have been widely considered. To know the relationship between the behavior of solutions and the geometry of domain, we showed both the relationship between the initial behavior and the curvatures of the boundary and that between the existence of a stationary level surface with time and the symmetry of domain. In particular, we obtained characterizations of the sphere, the hyperplane, and the circular cylinder involving a stationary level surface. These yielded a new development of inverse problems determining the geometry of domain. Also, as a by-product, we obtained Liouville-type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations describing an important class of Weingarten hypersurfaces.
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