Project/Area Number  15540150 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Akita University 
Principal Investigator 
KAWAKAMI Hajime Akita University, Faculty of Engineering and Resource Science, associate professor, 工学資源学部, 助教授 (20240781)

CoInvestigator(Kenkyūbuntansha) 
TSUCHIYA Masaaki Kanazawa University, School of Natural Science and Technology, professor, 大学院・自然科学研究科, 教授 (50016101)
KOBAYASHI Mahito Akita University, Faculty of Engineering and associate, associate professor, 工学資源学部, 助教授 (10261645)
SAKA Koichi Akita University, Faculty of Engineering and Resource Science, professor, 工学資源学部, 教授 (20006597)
MIKAMI Kentaro Akita University, Faculty of Engineering and Resource Science, professor, 工学資源学部, 教授 (70006592)

Project Period (FY) 
2003 – 2004

Project Status 
Completed(Fiscal Year 2004)

Budget Amount *help 
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 2004 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2003 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  inverse problem / heat equation / diffusion equation / shape of domain / Lipschitz domain / Dirichlet to Neumann map / mixed boundary value problem / 領域形状推定 / 混合型境界条件 / エネルギー評価 / 線形化 
Research Abstract 
The theme of our research is an inverse problem of determining the shape of some unknown portion of the boundary of a domain from measurements of some data for a solution to the heat equation on the domain. BryanCaudill [BC] (Inverse Problems, 1998) treated the problem of the case of Neumann boundary condition. And later, Y.Moriyama [M] (master thesis of Kanazawa Univ., 2002) studied the case of a mixed boundary condition. This condition fits physical phenomena well, and we also study the case of this condition. In [BC] and [M], the domain isa rectangular solid, and the coefficients and free terms of the equation are constants. On the other hand, in our research, the domain is a direct product of a Lipschitz domain and an interval, and the coefficients and free terms of the equation are Lipschitz continuous functions. The results of our research are as follows : (a)Based on [M], we got an energy estimate of weak solutions of a mixed boundary value problem of a heat equation on a Lipschitz domain. (b)In the same way as [BC] and [M], we linearized the inverse problem. We carried out it in a framework of weak forms, and gave the linearization by the Gateaux derivative. This enables us to treat the case of Lipschitz domains. In the proof we used the energy estimate above. (c)For the resulting linear problem above, we proved the reconstruction theorem which claims that we can uniquely determine unknown shape from given data. We showed some denseness of the range of a Dirichlet to Neumann map, and use it in the proof of the reconstruction theorem. (d)Under some assumption of unknown shape, we gave a method of approximate reconstruction of unknown shape and an estimation of the error.
