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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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Dirichlet problem in an angular domain with rapidly oscillating boundary: Modeling of the problem and asymptotics of the solution
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by S. A. Nazarov
Translated by: A. Plotkin
St. Petersburg Math. J. 19 (2008), 297-326
DOI: https://doi.org/10.1090/S1061-0022-08-01000-5
Published electronically: February 7, 2008

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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Bibliographic Information
  • S. A. Nazarov
  • Affiliation: Institute of Mechanical Engineering Problems, Bol’shoi Prospekt V.O. 61, St. Petersburg 199178, Russia
  • MR Author ID: 196508
  • Email: serna@snark.ipme.ru
  • Received by editor(s): October 10, 2006
  • Published electronically: February 7, 2008
  • Additional Notes: Supported by RFBR (grant 06-01-257)
  • © Copyright 2008 American Mathematical Society
  • 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
  • MathSciNet review: 2333903