Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators
Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators
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DOI：
10.1007/s00208-012-0786-0
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发表时间：
2013-02
影响因子：
1.4
通讯作者：
N. Honda;G. Nakamura;M. Sini
中科院分区：
文献类型：
--
作者：
N. Honda;G. Nakamura;M. Sini
研究了一类二阶标量椭圆算子的反障碍问题，该算子在障碍物附近具有实主部分和解析系数。我们假设障碍物的边界为非解析曲面。我们证明，当我们设置狄利克雷边界条件时，一次测量就足以重建障碍物。在诺伊曼情况下，我们得到的结果一般只有= 2,3。更准确地说，我们表明，一个测量是足够的形式= 2，我们需要3个线性无关的输入形式= 3。然而，对于亥姆霍兹方程，对于任意≥2，我们只需要- 1个线性无关输入。这里是包含障碍物的空间维度。通过研究实解析函数的零集的解析性，证明了这一点。此外，我们还给出了每一种情况下障碍物形状的重建步骤。虽然我们陈述了散射问题的结果，但类似的结果也适用于相关的边值问题。
We deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part andanalytic coefficientsnear the obstacle. We assume that the boundary of the obstacle is anon-analytichypersurface. We show that, when we put Dirichlet boundary conditions, one measurement is enough to reconstruct the obstacle. In the Neumann case, we have results only forn= 2, 3 in general. More precisely, we show that one measurement is enough forn= 2 and we need 3 linearly independent inputs forn= 3. However, in the case for the Helmholtz equation, we only needn− 1 linearly independent inputs, for anyn≥ 2. Herenis the dimension of the space containing the obstacle. These are justified by investigating the analyticity properties of the zero set of a real analytic function. In addition, we give a reconstruction procedure for each case to recover the shape of obstacle. Although we state the results for the scattering problems, similar results are true for the associated boundary value problems.