Ilia V. Kamotski

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The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. This may be seen for example as a simplified scalar cross-sectional model of the problem for localised modes in photonic crystal fibers. For a high contrast small size periodicity and a finite size defect we consider the critical (the so called double poros-ity(More)
We prove that the standard second-kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e. sign-definite) for all smooth convex domains when the wavenumber k is sufficient large. (This integral equation involves the so-called “combined potential” or “combined field” operator.) This coercivity result(More)
Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain nonstandard self-adjoint operators. In particular they(More)
Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral operators. It is shown that this series is an asymptotic expansion with respect to smoothness under quite general geometric(More)
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