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Mat. Zametki, 1975, Volume 17, Issue 1, Pages 91–101 (Mi mz7227)  

This article is cited in 11 scientific papers (total in 11 papers)

Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation

V. A. Il'in

V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, USSR

Abstract: In this article a uniqueness theorem for the classical solution of a mixed problem is proved under minimal assumptions on the coefficients of the differential operator for admitting the Fourier method of a hyperbolic second-order equation in an $(N+1)$-dimensional cylinder, whose cross section is a completely arbitrary bounded $N$-dimensional domain. Furthermore, it is proved that the classical solution of the indicated mixed problem, whenever it exists, belongs to the class $W^1_2$ and is the generalized solution from $W^1_2$ of the same problem.

Full text: PDF file (796 kB)

English version:
Mathematical Notes, 1975, 17:1, 53–58

Bibliographic databases:

UDC: 517.43
Received: 12.09.1974

Citation: V. A. Il'in, “Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation”, Mat. Zametki, 17:1 (1975), 91–101; Math. Notes, 17:1 (1975), 53–58

Citation in format AMSBIB
\Bibitem{Ili75}
\by V.~A.~Il'in
\paper Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation
\jour Mat. Zametki
\yr 1975
\vol 17
\issue 1
\pages 91--101
\mathnet{http://mi.mathnet.ru/mz7227}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=369929}
\zmath{https://zbmath.org/?q=an:0315.35055}
\transl
\jour Math. Notes
\yr 1975
\vol 17
\issue 1
\pages 53--58
\crossref{https://doi.org/10.1007/BF01093843}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. K. B. Sabitov, I. A. Khadzhi, “The boundary-value problem for the Lavrent'ev–Bitsadze equation with unknown right-hand side”, Russian Math. (Iz. VUZ), 55:5 (2011), 35–42  mathnet  crossref  mathscinet
    2. Sabitov K.B., “Initial-Boundary Value Problem for a Parabolic-Hyperbolic Equation with Power-Law Degeneration on the Type Change Line”, Differ. Equ., 47:10 (2011), 1490–1497  crossref  isi
    3. Sabitov K.B., “Nachalno-granichnaya zadacha dlya parabolo-giperbolicheskogo uravneniya so stepennym vyrozhdeniem na perekhodnoi linii”, Differentsialnye uravneniya, 47:10 (2011), 1474–1481  elib
    4. Sabitov K.B., “Zadacha dirikhle dlya uravneniya smeshannogo tipa tretego poryadka v pryamougolnoi oblasti”, Differentsialnye uravneniya, 47:5 (2011), 705–713  elib
    5. I. A. Khadzhi, “Inverse Problem for Equations of Mixed Type with Lavrentev–Bitsadze Operator”, Math. Notes, 91:6 (2012), 857–867  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. Sabitov K.B., “Kraevaya zadacha dlya uravneniya smeshannogo tipa tretego poryadka v pryamougolnoi oblasti”, Differentsialnye uravneniya, 49:2 (2013), 186–186  elib
    7. Sabitov K.B., Vagapova E.V., “Zadacha dirikhle dlya uravneniya smeshannogo tipa s dvumya liniyami vyrozhdeniya v pryamougolnoi oblasti”, Differentsialnye uravneniya, 49:1 (2013), 68–68  elib
    8. K. B. Sabitov, “The Dirichlet Problem for Higher-Order Partial Differential Equations”, Math. Notes, 97:2 (2015), 255–267  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. N. V. Martem'yanova, “The Dirichlet problem for an equation of mixed elliptic-hyperbolic type with variable potential”, Russian Math. (Iz. VUZ), 59:11 (2015), 36–44  mathnet  crossref
    10. A. A. Gimaltdinova, “Zadacha Dirikhle dlya uravneniya smeshannogo tipa s dvumya liniyami perekhoda v pryamougolnoi oblasti”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:4 (2015), 634–649  mathnet  crossref  zmath  elib
    11. A. A. Gimaltdinova, K. V. Kurman, “Boundary problem for Lavrent'ev–Bitsadze equation with two internal lines of change of a type”, Russian Math. (Iz. VUZ), 60:3 (2016), 18–31  mathnet  crossref  isi
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