Reconstruction of the Defect by the Enclosure Method for Inverse Problems of the Magnetic Schrödinger Operator

Pub Date : 2022-01-01 DOI:10.3836/tjm/1502179363
K. Kurata, Ryusei Yamashita
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引用次数: 1

Abstract

This study is based on the paper [1]. We give the formula to extract the position and the shape of the defect D generated in the object (conductor) Ω from the observation data on the boundary ∂Ω for the magnetic Schrödinger operator by using the enclosure method proposed by Ikehata [2]. We show a reconstruction formula of the convex hull of the defect D from the observed data, assuming certain higher regularity for the potentials of the magnetic Schrödinger operator, under the Dirichlet condition or the Robin condition on the boundary ∂D in the two and three dimensional case. Let Ω ⊂ R(n = 2, 3) be a bounded domain where the boundary ∂Ω is C and let D be an open set satisfying D ⊂ Ω and Ω \ D is connected. The defect D consists of the union of disjoint bounded domains {Dj}j=1, where the boundary of D is Lipschitz continuous. First, we define the DN map for the magnetic Schrödinger equation with no defect D in Ω. Here, let D Au := ∑n j=1 DA,j(DA,ju), where DA,j := 1 i ∂j +Aj and A = (A1, A2, · · · , An). Definition 1. Suppose q ∈ L∞(Ω), q ≥ 0, A ∈ C(Ω, R). For a given f ∈ H(∂Ω), we say u ∈ H(Ω) is a weak solution to the following boundary value problem for the magnetic Schrödinger equation { D Au+ qu = 0 in Ω, u = f on ∂Ω, (1.1)
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磁性Schrödinger算子逆问题的包体法缺陷重构
本研究基于论文[1]。利用Ikehata[2]提出的封闭方法,我们给出了从磁薛定谔算子边界上的观测数据中提取物体(导体)Ω中产生的缺陷D的位置和形状的公式。我们从观测数据中给出了缺陷D的凸包的重建公式,假设磁薛定谔算子的势在二维和三维情况下,在边界上的Dirichlet条件或Robin条件下具有更高的正则性。设Ω⊂R(n=2,3)是一个边界为C的有界域,设D是一个满足D⊁Ω的开集,Ω\D是连通的。缺陷D由不相交的有界域的并集组成{Dj}j=1,其中D的边界是Lipschitz连续的。首先,我们定义了Ω中没有缺陷D的磁薛定谔方程的DN映射。这里,设D Au:=∑n j=1 DA,j(DA,ju),其中DA,j:=1 iõj+Aj和A=(A1,A2,··,An)。定义1。设q∈L∞(Ω),q≥0,A∈C(Ω,R)。对于给定的f∈H(⏴Ω),我们认为u∈H是磁薛定谔方程{D Au+qu=0 inΩ,u=f on⏴Ω,(1.1)的边值问题的弱解
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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