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

This article is cited in 13 scientific papers (total in 13 papers)

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
&\qquad=f(x)-\operatorname{div} F(x), \qquad x\in Q, \end{split}
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.


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

Bibliographic databases:

UDC: 517.956
MSC: 35J15
Received: 08.11.2010

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
\by V.~Zh.~Dumanyan
\paper Solvability of the Dirichlet problem for a~general second-order elliptic equation
\jour Mat. Sb.
\yr 2011
\vol 202
\issue 7
\pages 75--94
\jour Sb. Math.
\yr 2011
\vol 202
\issue 7
\pages 1001--1020

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    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  mathnet
    5. V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theoret. and Math. Phys., 180:2 (2014), 917–931  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    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  mathnet  crossref  zmath  elib
    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  mathnet  crossref  crossref  isi  elib
    8. A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    11. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    12. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    13. A. K. Gushchin, “Extensions of the space of continuous functions and embedding theorems”, Sb. Math., 211:11 (2020), 1551–1567  mathnet  crossref  crossref  mathscinet  isi  elib
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