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This article is cited in 4 scientific papers (total in 4 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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Article 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. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43
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A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439
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A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839
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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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