2015 Fiscal Year Final Research Report
Search for new isoperimetric inequalities relating to elliptic equations
| Project/Area Number |
25610024
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
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| Co-Investigator(Renkei-kenkyūsha) |
JIMBO SHUICHI 北海道大学, 大学院理学研究院, 教授 (80201565)
SHIBATA TETSUTARO 広島大学, 大学院工学研究院, 教授 (90216010)
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| Project Period (FY) |
2013-04-01 – 2016-03-31
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| Keywords | 楕円型境界値問題 / 等距離摂動領域 / 球対称性 / 等位集合 / 幾何学的特性 / 強比較定理 / アレクサンドロフの反射原理 |
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| Outline of Final Research Achievements |
We show that if a unique solution of an elliptic boundary value problem over a bounded domain has a level surface parallel to the boundary, then the domain must be a ball. This is regarded as a kind of isoperimetric property such that the rate of change of the solution on the surface parallel to the boundary must be minimized at a ball. Moreover, it is shown that, if the solution is nearly a constant on the surface, then the domain is nearly a ball. Also, we show that if a unique solution of an elliptic boundary value problem on an exterior domain has a level surface similar to the boundary, then the boundary must be a sphere. On the other hand, for a hypersurface, we consider the perturbed domain whose boundary consists of two hypersurfaces parallel to the hypersurface at an equal distance. Under some initial and boundary conditions, we show that if the hypersurface is stationary isotermic, then the surface must be either a plane or a sphere. Some new problems are proposed.
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| Free Research Field |
偏微分方程式論
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