Existence, Uniqueness and Regularity for Solutions of the Conical Diffraction Problem
Authors
- Elschner, Johannes
- Hinder, Rainer
- Penzel, Frank
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 78A45 35J50 35J20 35B65
Keywords
- Maxwell equations, system of Helmholtz equations, transmission problem, strongly elliptic variational formulation, singularities of solutions
DOI
Abstract
This paper is devoted to the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasi-periodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.
Appeared in
- Math. Models Meth. Appl. Sci. 10 (2000) pp. 317-341
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