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Dirichlet problem in an angular domain with rapidly oscillating boundary: Modeling of the problem and asymptotics of the solution


Author: S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 19 (2007), nomer 2.
Journal: St. Petersburg Math. J. 19 (2008), 297-326
MSC (2000): Primary 35B40, 35J25
DOI: https://doi.org/10.1090/S1061-0022-08-01000-5
Published electronically: February 7, 2008
MathSciNet review: 2333903
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Abstract | References | Similar Articles | Additional Information

Abstract: Leading asymptotic terms are constructed and justified for the solution of the Dirichlet problem corresponding to the Poisson equation in an angular domain with rapidly oscillating boundary. In addition to an exponential boundary layer near the entire boundary, a power-law boundary layer arises, which is localized in the vicinity of the corner point. Modeling of the problem in a singularly perturbed domain is studied; this amounts to finding a boundary-value problem in a simpler domain whose solution approximates that of the initial problem with advanced precision, namely, yields a two-term asymptotic expression. The way of modeling depends on the opening $ \alpha$ of the angle at the corner point; the cases where $ \alpha<\pi$, $ \alpha\in(\pi,2\pi)$, and $ \alpha=2\pi$ are treated differently, and some of them require the techniques of selfadjoint extensions of differential operators.


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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol’shoi Prospekt V.O. 61, St. Petersburg 199178, Russia
Email: serna@snark.ipme.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01000-5
Keywords: Dirichlet problem, oscillating boundary, corner point, asymptotics, selfadjoint extension
Received by editor(s): October 10, 2006
Published electronically: February 7, 2008
Additional Notes: Supported by RFBR (grant 06-01-257)
Article copyright: © Copyright 2008 American Mathematical Society