2006 Fiscal Year Final Research Report Summary
Behavior of spatial critical points and level surfaces of solutions of partial differential equations and shapes of the solutions
| Project/Area Number |
15340047
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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 | Ehime University |
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Principal Investigator |
SAKAGUCHI Shigeru Ehime University, Graduate School of Science and Engineering, Professor, 理学部, 教授 (50215620)
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| Co-Investigator(Kenkyū-buntansha) |
MIKAMI Toshio Hokkaido University, Graduate School of Science, Associate Professor, 理学研究院, 助教授 (70229657)
HASHIMOTO Takahiro Ehime University, Graduate School of Science and Engineering, Research Associate, 理工学研究科, 助手 (60291499)
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| Project Period (FY) |
2003 – 2006
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| Keywords | diffusion equation / nonlinear diffusion equation / initial-boundary value problem / initial value problem / isothermic surface / hot spot / heat equation / level surface |
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| Research Abstract |
The main purpose of this project is to study the relationship between shape of solutions of partial differential equations (behavior of spatial critical points and level surfaces) and shape of domains. We obtain the following:
1. Consider the initial value problem for the heat equation in Euclidean space with initial data being the characteristic function of a domain Ω. We introduce the geometrical condition that Ω is uniformly dense in Γ, which is necessary for the solution to have a stationary isothermic surface Γ. The uniformly dense domains are classified. (Trans. Amer. Math. Soc., 358 (2006), 4821-4841 )
2. Consider the initial-boundary value problem for linear and nonlinear diffusion equations in a bounded domain Ω in Euclidean space with zero initial data and with positive constant boundary value. Let B be a ball in Ω touching ∂ Ω only at one point. Then the asymptotic formula of the integral of the solution over B at the initial time involves the principal curvatures of ∂ Ω at th
… More
e point. This fact explains the relationship between diffusion and the geometry of Ω. ( Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 373-388 )
3. Consider the initial-Dirichlet problem for the heat equation with positive constant initial data in a domain Ω with unbounded boundary ∂ Ω in Euclidean space. Under various global assumptions on Ω, we prove that if the solution has a stationary isothermic surface, then ∂ Ω consists of hyperplanes. ( Indiana University Math. J., to appear )
4. Consider the initial-Dirichlet problem for the heat equation with positive constant initial data over a bounded convex polygonal domain Ω in the plane. When Ω has m (m≦5) sides and every side of ∂ Ω touches the inscribed circle, we obtain a new necessary condition for Ω having a stationary hot spot. (submitted for the publication )
5. In the initial-boundary value problem for the linear diffusion equation with zero initial data and with positive constant boundary value, a result of Varadhan (1967) is such that the initial behavior of the solution is described through the distance function to the boundary. We extend this result to some nonlinear diffusion equations, which are uniformly parabolic, with the aid of the theory of viscosity solutions. Moreover, we give a characterization of the sphere through the solution having a stationary level surface in case of nonlinear diffusion equations. (in preparation ) Less
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Research Products
(6 results)
