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 Mat. Sb., 2011, Volume 202, Number 7, Pages 75–94 (Mi msb7814)

Solvability of the Dirichlet problem for a general second-order elliptic equation

V. Zh. Dumanyan

Yerevan State University

Abstract: The paper is concerned with studying the solvability of the Dirichlet problem for the second-order elliptic equation
\begin{gather*} \begin{split} & -\operatorname{div} (A(x)\nabla u)+(\overline b(x),\nabla u)-\operatorname{div} (\overline c(x)u)+d(x)u
u|_{\partial Q}=u_0, \end{gather*}
in a bounded domain $Q\subset R_n$, $n\ge 2$, with $C^1$-smooth boundary and boundary condition $u_0\in L_2(\partial Q)$.
Conditions for the existence of an $(n-1)$-dimensionally continuous solution are obtained, the resulting solvability condition is shown to be similar in form to the solvability condition in the conventional generalized setting (in $W_2^1(Q)$). In particular, the problem is shown to have an $(n-1)$-dimensionally continuous solution for all $u_0\in L_2(\partial Q)$ and all $f$ and $F$ from the appropriate function spaces, provided that the homogeneous problem (with zero boundary conditions and zero right-hand side) has no nonzero solutions in $W_2^1(Q)$.
Bibliography: 14 titles.

Keywords: Dirichlet problem, solvability of the Dirichlet problem, second-order elliptic equation, $(n-1)$-dimensionally continuous solution.

DOI: https://doi.org/10.4213/sm7814

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English version:
Sbornik: Mathematics, 2011, 202:7, 1001–1020

Bibliographic databases:

Document Type: Article
UDC: 517.956
MSC: 35J15

Citation: V. Zh. Dumanyan, “Solvability of the Dirichlet problem for a general second-order elliptic equation”, Mat. Sb., 202:7 (2011), 75–94; Sb. Math., 202:7 (2011), 1001–1020

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7814
• https://doi.org/10.4213/sm7814
• http://mi.mathnet.ru/eng/msb/v202/i7/p75

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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. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Sb. Math., 203:1 (2012), 1–27
2. A. K. Guschin, “$L_p$-otsenki nekasatelnoi maksimalnoi funktsii dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 53–69
3. A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219
4. V. Zh. Dumanyan, “On boundedness of a class of first order linear differential operators in the space of $(n-1)$-dimensionally continuous functions”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 8–14
5. V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theoret. and Math. Phys., 180:2 (2014), 917–931
6. A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43
7. A. K. Gushchin, “V.A. Steklov's work on equations of mathematical physics and development of his results in this field”, Proc. Steklov Inst. Math., 289 (2015), 134–151
8. A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439
9. 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
10. 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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