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Research on an estimation problem for the shape of time-varying domain via parabolic equations

时变域形状的抛物方程估计问题研究

基本信息

批准号：
18540155
负责人：
KAWAKAMI Hajime
金额：
$ 1.09万
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：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540155/
关键词：
inverse problem identification of shape parabolic equation heat equation mixed boundary condition 未知形状

项目摘要

Our research program is concerned with an inverse problem of determining the shape of some time-varying unknown portion of the boundary of a multi-dimensional domain via a parabolic equation on the domain. Considering practical applications, we treat such a domain under weak regularity conditions and a parabolic operator of general type, and set a mixed boundary condition: Dirichlet's condition is imposed on the unknown portion and Robin's one on the other portions.As an observable data, we take the boundary value on an accessible portion of the boundary of a solution to the parabolic equation. The correspondence between data and domains is generally nonlinear. Thus we first considered a linearized problem; then, for the shape of the unknown portion, we proved a unique identification theorem and provided a reconstruction algorithm from the data We also verified the convergence and stability of the algorithm. The result was reported in the symposium "Inverse Problems in Applied Sciences -towards breakthrough-", and published in the journal " Inverse Problems".In the last term of the project, we considered the primary (not linearized) problem and obtained a unique identification theorem as follows. We first assume that the domain is Lipschitz and that the parabolic operator is non-degenerate and has bounded Lipschitz continuous coefficients. Secondarily we assume that the Robin boundary value does not vanish somewhere on the accessible portion at every observation time. Moreover we assume that the shape of the domain is known at the initial observation time or that the initial value of the solution is zero. Then, in the observation period, the shape of the unknown portion is uniquely determined by the data. The result was reported in the Annual Meeting of the Mathematical Society of Japan this spring. We are preparing to publish the details of the result.

我们的研究程序涉及通过区域上的抛物方程来确定多维区域的边界的某个时变未知部分的形状的反问题。考虑到实际应用，我们在弱正则性条件下和一般类型的抛物型算子下处理这类区域，并设置混合边界条件：在未知部分施加Dirichlet条件，在另一部分施加Robin条件，作为可观测数据，取抛物方程解的边界的可达部分的边值。数据和域之间的对应关系通常是非线性的。因此，我们首先考虑了一个线性化问题；然后，对于未知部分的形状，我们证明了一个唯一的识别定理，并给出了一个由数据重建的算法，并验证了算法的收敛和稳定性。这一结果发表在《应用科学中的逆问题--走向突破》研讨会上，并发表在《逆问题》杂志上。在该项目的最后一学期，我们考虑了主要的(未线性化的)问题，得到了一个唯一的识别定理如下。我们首先假设区域是Lipschitz的，抛物算子是非退化的，并且具有有界的Lipschitz连续系数。其次，我们假设Robin边值不会在每个观测时刻都消失在可达部分的某处。此外，我们假设在初始观测时区域的形状是已知的，或者解的初值为零。然后，在观察期内，由数据唯一地确定未知部分的形状。这一结果发表在今年春天召开的日本数学学会年会上。我们正准备公布结果的细节。