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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2020, Volume 17, Pages 2142–2189
DOI: https://doi.org/10.33048/semi.2020.17.144
(Mi semr1338)
 

This article is cited in 1 scientific paper (total in 1 paper)

Real, complex and functional analysis

Criterion for the Sobolev well-posedness of the Dirichlet problem for the Poisson equation in Lipschitz domains. I

A. I. Parfenov

Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
Full-text PDF (695 kB) Citations (1)
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Abstract: We study the Dirichlet problem for the Poisson equation in bounded Lipschitz domains. We show that its well-posedness in the first order Sobolev space is equivalent to the condition of K. Nyström (1996). This criterion is simpler than the similar criterion of Z. Shen (2005) due to using one positive harmonic function with vanishing trace instead of gradients of all harmonic functions with vanishing trace. Our criterion yields the main known facts about this well-posedness except for Shen's criterion. Finally, we determine all possible combinations of three basic properties (injectivity, denseness of range and closedness of range) of the operator of the boundary value problem under consideration.
Keywords: Alkhutov criterion, Bogdan formula for the Green function, Carleman–Huber theorem, Dirichlet problem for the Poisson equation, LHMD property, Lipschitz domain, Nyström condition, Shen criterion.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 0314-2019-0010
Received August 31, 2020, published December 22, 2020
Bibliographic databases:
Document Type: Article
UDC: 517.956.225
MSC: 35J05
Language: Russian
Citation: A. I. Parfenov, “Criterion for the Sobolev well-posedness of the Dirichlet problem for the Poisson equation in Lipschitz domains. I”, Sib. Èlektron. Mat. Izv., 17 (2020), 2142–2189
Citation in format AMSBIB
\Bibitem{Par20}
\by A.~I.~Parfenov
\paper Criterion for the Sobolev well-posedness of the Dirichlet problem for the Poisson equation in Lipschitz domains.~I
\jour Sib. \`Elektron. Mat. Izv.
\yr 2020
\vol 17
\pages 2142--2189
\mathnet{http://mi.mathnet.ru/semr1338}
\crossref{https://doi.org/10.33048/semi.2020.17.144}
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