Article

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

Integral Equations and Operator Theory (Impact Factor: 0.58). 05/2011; 73(1). DOI: 10.1007/s00020-012-1955-y
Source: arXiv

ABSTRACT We develop the shape derivative analysis of solutions to the problem of
scattering of time-harmonic electromagnetic waves by a bounded penetrable
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 $TH\sp{-1/2}(\Div_{\Gamma},\Gamma)$. 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 G\^ateaux
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.

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Available from: Martin Costabel, Feb 19, 2014
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    • "For this, the tools presented above are not directly applicable. It is the purpose of the second part [3] of our paper to present an alternative strategy using the Helmholtz decomposition of the space TH − 1 2 (div Γ , Γ). "
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