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 TMF, 2014, Volume 180, Number 2, Pages 189–205 (Mi tmf8670)

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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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/tmf8670
• https://doi.org/10.4213/tmf8670
• http://mi.mathnet.ru/eng/tmf/v180/i2/p189

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43
2. A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439
3. 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
4. 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
5. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
6. A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65
7. A. K. Guschin, “Obobscheniya prostranstva nepreryvnykh funktsii; teoremy vlozheniya”, Matem. sb., 211:11 (2020), 54–71
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