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June  2015, 8(3): 435-473. doi: 10.3934/dcdss.2015.8.435

## Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures

 1 Berlin Mathematical School, Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Received  November 2013 Revised  March 2014 Published  October 2014

This paper is concerned with the study of a new integral equation formulation for electromagnetic scattering by a $2\pi$-biperiodic polyhedral Lipschitz profile. Using a combined potential ansatz, we derive a singular integral equation with Fredholm operator of index zero from time-harmonic Maxwell's equations and prove its equivalence to the electromagnetic scattering problem. Moreover, under certain assumptions on the electric permittivity and the magnetic permeability, we obtain existence and uniqueness results in the special case that the grating is smooth and, under more restrictive assumptions, in the case that the grating is of polyhedral Lipschitz regularity.
Citation: Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435
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