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 Mat. Tr., 2017, Volume 20, Number 1, Pages 158–200 (Mi mt320)

Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem

A. I. Parfenov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: In a special Lipschitz domain treated as a perturbation of the upper half-space, we construct a perturbation theory series for a positive harmonic function with zero trace. The terms of the series are harmonic extensions to the half-space from its boundary of distributions defined by a recurrent formula and passage to the limit. The approximation error by a segment of the series is estimated via a power of the seminorm of the perturbation in the homogeneous Slobodestkiĭ space $b_N^{1-1/N}$. The series converges if the Lipschitz constant of the perturbation is small.

Key words: positive harmonic function, conformal mapping, Lipschitz continuous perturbation of the boundary.

DOI: https://doi.org/10.17377/mattrudy.2017.20.110

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English version:
Siberian Advances in Mathematics, 2017, 27:4, 274–304

Document Type: Article
UDC: 517.572

Citation: A. I. Parfenov, “Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem”, Mat. Tr., 20:1 (2017), 158–200; Siberian Adv. Math., 27:4 (2017), 274–304

Citation in format AMSBIB
\Bibitem{Par17} \by A.~I.~Parfenov \paper Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem \jour Mat. Tr. \yr 2017 \vol 20 \issue 1 \pages 158--200 \mathnet{http://mi.mathnet.ru/mt320} \crossref{https://doi.org/10.17377/mattrudy.2017.20.110} \elib{http://elibrary.ru/item.asp?id=29145408} \transl \jour Siberian Adv. Math. \yr 2017 \vol 27 \issue 4 \pages 274--304 \crossref{https://doi.org/10.3103/S1055134417040058} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85036570782} 

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This publication is cited in the following articles:
1. A. I. Parfënov, “Priblizhennoe vychislenie defekta lipshitseva tsilindricheskogo kondensatora”, Sib. elektron. matem. izv., 15 (2018), 906–926
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