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This article is cited in 7 scientific papers (total in 7 papers)
Solvability of the Dirichlet problem for second-order elliptic equations
V. Zh. Dumanyan
Yerevan State University, Yerevan, Armenia
Abstract:
In our preceding papers, we obtained necessary and sufficient conditions for the existence of an $(n{-}1)$-dimensionally continuous solution of the Dirichlet problem in a bounded domain $Q\subset\mathbb R_n$ under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space $W_2^1(Q)$ under the additional requirement that the boundary function belongs to the space $W_2^{1/2}(\partial Q)$.
Keywords:
Dirichlet problem, elliptic equation
DOI:
https://doi.org/10.4213/tmf8670
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English version:
Theoretical and Mathematical Physics, 2014, 180:2, 917–931
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Received: 28.02.2014
Revised: 27.03.2014
Citation:
V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, TMF, 180:2 (2014), 189–205; Theoret. and Math. Phys., 180:2 (2014), 917–931
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/tmf8670https://doi.org/10.4213/tmf8670 http://mi.mathnet.ru/eng/tmf/v180/i2/p189
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A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64
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A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
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