An estimation problem for the shape of a domain via diffusion equations

通过扩散方程估计域形状的问题

• 批准号: 15540150
• 负责人: KAWAKAMI Hajime
• 金额: $ 0.96万
• 依托单位: Akita University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540150/
• 关键词: 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

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. Bryan-Caudill [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 is-a 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.

我们的研究的主题是一个反问题，确定的形状的一些未知部分的边界的一个域从测量的一些数据的解决方案的热方程的域。Bryan-Caudill [BC]（InverseProblems，1998）处理了Neumann边界条件情形的问题。后来，Y.Moriyama [M]（金泽大学硕士论文，2002）研究了混合边界条件的情况。这个条件很好地符合物理现象，我们也研究了这个条件的情况。在[BC]和[M]中，区域是长方体，方程的系数和自由项是常数。另一方面，在我们的研究中，区域是Lipschitz区域和区间的直积，并且方程的系数和自由项是Lipschitz连续函数。（a）在[M]的基础上，得到了Lipschitz区域上一类热传导方程混合边值问题弱解的能量估计。(b)In与[BC]和[M]一样，我们将逆问题线性化。我们在弱形式的框架下实现了它，并给出了线性化的Gateaux导数。这使我们能够处理Lipschitz域的情况。在证明中，我们使用了上面的能量估计。（c）对于上述线性问题，我们证明了重构定理，该定理声称我们可以从给定的数据中唯一地确定未知形状。我们证明了Dirichlet到Neumann映射值域的稠密性，并将其用于重构定理的证明。（4）在未知形状的假设下，给出了一种近似重构未知形状的方法，并给出了误差估计。