Revisiting the probe and enclosure methods
Revisiting the probe and enclosure methods
复制标题
重新审视探头和外壳方法
DOI：
10.1088/1361-6420/ac70f2
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发表时间：
2022
期刊：
影响因子：
2.1
通讯作者：
Masaru Ikehata
中科院分区：
文献类型：
--
作者：
Kousuke Kuto;Tatsuki Mori;Tohru Tsujikawa;Shoji Yotsutani;J. Sugie;Masaru Ikehata
本文关注的是偏微分方程控制的逆障碍问题的重构问题，由两部分组成。(i)第一部分考虑了固定薛定谔方程控制的嵌入介质中的不可穿透障碍物的探测和封闭方法的基础。在一般框架下，给出了根据介质所占据的物体外表面的一对狄利克雷和诺依曼数据计算出的量的一些自然估计。这些估计使我们能够几乎立即推导出这两种方法的指示函数的必要渐近行为。(ii)第二个方法考虑了嵌入在由稳态薛定谔方程控制的吸收介质中的可穿透障碍物的封闭方法的实现。这里考虑的未知障碍物是通过添加到背景复值 L∞ 势的扰动项来建模的。在跨越障碍物边界项的跳跃条件和障碍物表面的某种规律性（包括利普希茨（Lipschitz））作为特例的情况下，建立了使用无限多对狄利克雷和诺伊曼数据的包围方法。
This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts.(i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schrödinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behaviour of indicator functions for both methods.(ii) The second one considers the realization of the enclosure method for a penetrable obstacle embedded in an absorbing medium governed by the stationary Schrödinger equation. The unknown obstacle considered here is modeled by a perturbation term added to the background complex-valued L∞-potential. Under a jump condition on the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established.