Probe method and a Carleman function
Probe method and a Carleman function
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DOI：
10.1088/0266-5611/23/5/006
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发表时间：
2007-01
期刊：
影响因子：
2.1
通讯作者：
Masaru Ikehata
中科院分区：
文献类型：
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作者：
Masaru Ikehata
Carleman函数是拉普拉斯算子的一种特殊的大参数基本解，它给出了计算拉普拉斯方程柯西问题在一个区域上的解的值的公式.有界区域上拉普拉斯方程反边值问题的探针法是基于调和函数列的存在性，这种调和函数列称为针列。Needle序列在一条特殊曲线上爆炸，该曲线连接域内的一个给定点和域边界上的一个点，并且在曲线外局部收敛。序列产生一个未知的不连续性，如一个腔，包含在一个给定的介质从狄利克雷到诺依曼映射的重建公式。本文以封闭形式给出了一个三维显式needle序列。这是一个应用程序的Carleman函数介绍了Yarmukhamedov。此外，还给出了将具有任意固定波数的障碍物散射反问题化为Helmholtz方程边值反问题的显式探针方法。
A Carleman function is a special fundamental solution with a large parameter for the Laplace operator and gives a formula to calculate the value of the solution of the Cauchy problem in a domain for the Laplace equation. The probe method applied to an inverse boundary value problem for the Laplace equation in a bounded domain is based on the existence of a special sequence of harmonic functions which is called a needle sequence. The needle sequence blows up on a special curve which connects a given point inside the domain with a point on the boundary of the domain and is convergent locally outside the curve. The sequence yields a reconstruction formula of an unknown discontinuity, such as a cavity, inclusion in a given medium from the Dirichlet-to-Neumann map. In this paper, an explicit needle sequence in three dimensions is given in closed form. It is an application of a Carleman function introduced by Yarmukhamedov. Furthermore, an explicit needle sequence in the probe method applied to the reduction of inverse obstacle scattering problems with an arbitrary fixed wave number to inverse boundary value problems for the Helmholtz equation is also given.