Uniqueness in shape identification of a time-varying domain and related parabolic equations on non-cylindrical domains
Uniqueness in shape identification of a time-varying domain and related parabolic equations on non-cylindrical domains
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时变域形状识别的唯一性及非圆柱域上的相关抛物线方程
DOI：
10.1088/0266-5611/26/12/125007
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发表时间：
2010
期刊：
影响因子：
2.1
通讯作者：
M. Tsuchiya
中科院分区：
文献类型：
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作者：
Hajime Kawakami;M. Tsuchiya
本文讨论了用温度边界测量法确定时变李普希茨域形状的反问题；这样的域被视为时空中的非圆柱形域。本文主要研究形状识别的唯一性。作为证明唯一性的一般处理，引入了一对域上的可比性条件；在时间无关的情况下，条件自动成立。在此基础上，我们给出了在适当的初始形状条件或初始温度条件下形状识别保持唯一性的几类域。每一类的每一个元素都有一个特定的几何条件；特别地，证明了多面体域类和任意一类具有C1光滑且具有共同初始形状的域满足唯一性。利用具有混合边界条件的抛物型方程研究了反问题。在非柱面非lipschitz区域上具有齐次Dirichlet边界条件的诱导抛物方程的弱解的惟一延拓性和弱解的惟一性发挥了关键作用。
The paper deals with an inverse problem determining the shape of a time-varying Lipschitz domain by boundary measurements of the temperature; such a domain is treated as a non-cylindrical domain in the time-space. Here we focus on the uniqueness of the shape identification. As a general treatment to show the uniqueness, a comparability condition on a pair of domains is introduced; the condition holds automatically in the time-independent case. Based on the condition, we provide several classes of domains in which the uniqueness of the shape identification holds under an appropriate initial shape condition or initial temperature condition. Each of such classes is characterized by a certain geometric condition on its each single element; in particular, it is verified that the class of polyhedral domains and any class of domains with C1 smoothness and with a common initial shape fulfil the uniqueness property. The inverse problem is studied via a parabolic equation with a mixed boundary condition. Then the unique continuation property of weak solutions and the uniqueness of weak solutions to an induced parabolic equation with the homogeneous Dirichlet boundary condition on a non-cylindrical non-Lipschitz domain play key roles.
通过抛物线方程求解域形状随时间变化的估计问题
DOI：
--
发表时间：
2007
期刊：
Inverse Problems 23
影响因子：
--
作者：
Hajime Kawakami;Yosuke Moriyama and Masaaki Tsuchiya
通讯作者：
Yosuke Moriyama and Masaaki Tsuchiya