Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle
Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle
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DOI：
10.1007/s00020-012-1955-y
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发表时间：
2011-05
影响因子：
0.8
通讯作者：
M. Costabel;F. Le Louër
中科院分区：
文献类型：
--
作者：
M. Costabel;F. Le Louër
我们开发了形状导数分析来解决时谐电磁波被可穿透有界障碍物散射的问题。由于边界积分方程是解决电磁散射问题的经典工具，因此我们研究了标准电磁边界积分算子的形状可微性质。后者通常受混合正则切向矢量场空间的限制。使用亥姆霍兹分解，我们可以将其分析基于标准 Sobolev 空间中的伪微分积分算子的研究，但随后我们必须研究曲面微分算子的 Gâteaux 可微性。我们证明了电磁边界积分算子是无限可微的，且不损失正则性。我们还描述了介电散射问题解的一阶形状导数作为新电磁散射问题的解。
We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.