Abstract

We consider the scattering problem governed by the Helmholtz equation with inhomogeneity in both “conductivity” in the divergence form and “potential” in the lower order term. The support of the inhomogeneity is assumed to contain a convex corner. We prove that, due to the presence of such corner under appropriate assumptions on the potential and conductivity in the vicinity of the corner, any incident field scatters. Based on corner scattering analysis we present a uniqueness result on determination of the polygonal convex hull of the support of admissible inhomogeneities, from scattering data corresponding to one single incident wave. These results require only certain regularity around the corner for the coefficients modeling the inhomogeneity, whereas away from the corner they can be quite general. Our main results on scattering and inverse scattering are established for ${\mathbb{R}}^2,$ while some analytic tools are developed in any dimension $n\ge 2.$

摘要

我们考虑由Helmholtz方程控制的散射问题，其发散形式的“电导率”和低阶项的“电势”均不均匀。假设不均匀性的支撑包含凸角。我们证明，由于在适当假设下在拐角附近的电势和电导率下存在该拐角，因此任何入射场都会发生散射。基于角散射分析，我们通过确定对应于单个入射波的散射数据，确定了可允许的不均匀性支撑的多边形凸包，从而给出了唯一性结果。这些结果只需要拐角处的一定规律性就可以对不均匀性进行建模，而远离拐角处的系数可能会很笼统。${\mathbb{[R}}^{\mathrm{2个}}，$ 而某些分析工具却在各个维度上得到了发展 $ñ\ge \mathrm{2个}。$