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

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

Bibliographic databases:

Document Type: 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

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  • 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  mathnet  crossref  zmath  elib
    2. 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
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    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  mathnet  crossref  crossref  isi  elib  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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