Neumann problem in angular regions with periodic and parabolic perturbations of the boundary

Nazarov, S.

doi:10.1090/S0077-1554-08-00173-8

Neumann problem in angular regions with periodic and parabolic perturbations of the boundary
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by S. A. Nazarov
Translated by: O. A. Khleborodova

Trans. Moscow Math. Soc. 2008, 153-208

DOI: https://doi.org/10.1090/S0077-1554-08-00173-8

Published electronically: December 24, 2008

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Abstract:

We construct and prove asymptotic expansions at infinity for solutions of Neumann problems and matching problems for systems of second order differential equations in regions with corner outlets to infinity. Outside some disc the regions are either periodic or deformed by parabolic inclusions. In addition to logarithmic-polynomial solutions, the asymptotic expansions contain components of the type of boundary layer that either exponentially decay away from the boundary or are localized inside the parabolic inclusions. Operators of the problems become Fredholm operators, and remainders in asymptotic expansions are estimated in scales of function spaces with norms determined by double weight factors and their step distribution. We consider also other types of problems which allow us to apply the developed methods for reduction to a model problem in a sector and for recovery of properties of remainders near perturbed boundary; in particular, we consider the matching problem in regions with irregular points of peak-like inclusion type.

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