In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in $ \Omega\setminus\overline D $ with an unknown obstacle $ D\Subset\Omega $ whose boundary $ \partial D $ is visible from the boundary $ \partial\Omega $ of $ \Omega $ and a measurement is given as a set of Cauchy data on $ \partial\Omega $. Here the Cauchy data is given by a unique solution $ u $ of the boundary value problem for the Laplace equation in $ \Omega\setminus\overline D $ with homogeneous and inhomogeneous Dirichlet boundary condition on $ \partial D $ and $ \partial\Omega $, respectively. These testing methods are domain sampling methods to estimate the location of the obstacle using test domains and the associated indicator functions. Also both of these testing methods can test the analytic extension of $ u $ to the exterior of a test domain. Since these methods are defined via some operators which are dual to each other, we could expect that there is a duality between the two methods. We will give this duality in terms of the equivalence of the pre-indicator functions associated to their indicator functions. As an application of the duality, the reconstruction of $ D $ using the RT gives the reconstruction of $ D $ using the NRT and vice versa. We will also give each of these reconstructions without using the duality if the Dirichlet data of the Cauchy data on $ \partial\Omega $ is not identically zero and the solution to the associated forward problem does not have any analytic extension across $ \partial D $. Moreover, we will show that these methods can still give the reconstruction of $ D $ if we a priori knows that $ D $ is a convex polygon and it satisfies one of the following two properties: all of its corner angles are irrational and its diameter is less than its distance to $ \partial\Omega $.

中文翻译：

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范围测试与无响应测试之间的对偶及其在反问题中的应用

在本文中，我们将说明$ \ Omega \ setminus \ overline D $中具有未知障碍物$ D \的Laplace方程的逆边值问题的范围测试（RT）和无响应测试（NRT）之间的对偶性子集\ Omega $，其边界$ \ partial D $从$ \ Omega $的边界$ \ partial \ Omega $中可见，并且以$ \ partial \ Omega $上的一组柯西数据给出度量。这里的柯西数据由唯一的解$ u $给出，其中$ \ Omega \ setminus \ overline D $的Laplace方程的边值问题具有$ \ partial D $和$ \ partial \的齐次和不齐次Dirichlet边界条件欧米茄$。这些测试方法是域采样方法，用于使用测试域和关联的指标函数来估计障碍物的位置。这两种测试方法还可以测试$ u $到测试域外部的分析扩展。由于这些方法是通过一些互为对偶的运算符定义的，因此我们可以期望这两种方法之间存在对偶性。我们将根据与它们的指标功能相关的前指标功能的等效性来给出这种二重性。作为对偶的应用，使用RT重建$ D $可以使用NRT重建$ D $，反之亦然。如果$ \ partial \ Omega $上的Cauchy数据的Dirichlet数据不完全为零，并且相关正向问题的解在$ \ partial D上没有任何解析扩展，我们还将在不使用对偶的情况下给出这些重构中的每一个。 $。此外，