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Mat. Sb. (N.S.), 1971, Volume 84(126), Number 1, Pages 27–65 (Mi msb3028)  

This article is cited in 10 scientific papers (total in 11 papers)

Boundary value problems for systems with a parameter

M. S. Agranovich

Abstract: The paper is concerned with problems of the form
$$ A(x,\frac\partial{\partial x},p)u(x)=f(x)\quadin\quad G,\qquad B(x,\frac\partial{\partial x},p)u(x)=g(x)\quadon\quad\Gamma. $$
Here $G$ is a region in $R_x^n$ with smooth boundary $\Gamma$; $A$ and $B$ are matrices of linear partial differential operators with smooth coefficients, depending polynomially on the complex parameter $p$. The operator $A$ is obtained by replacing $\partial/\partial x$ by $p$ in the operator $A(x,\partial/\partial x,\partial/\partial t)$, which is strongly hyperbolic in the sense of I. G. Petrovskii. Under some supplementary assumptions, the existence and uniqueness of a strong solution in the spaces $H_{qs}$ is demonstrated, and an a priori estimate in norms involving the parameter $p$ is obtained for large values $\operatorname{Re}p$.
Bibliography: 30 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 13:1, 25–64

Bibliographic databases:

UDC: 517.944.4
MSC: 35L50, 35L35, 35A05, 35B45, 35Sxx
Received: 29.05.1970

Citation: M. S. Agranovich, “Boundary value problems for systems with a parameter”, Mat. Sb. (N.S.), 84(126):1 (1971), 27–65; Math. USSR-Sb., 13:1 (1971), 25–64

Citation in format AMSBIB
\by M.~S.~Agranovich
\paper Boundary value problems for systems with a~parameter
\jour Mat. Sb. (N.S.)
\yr 1971
\vol 84(126)
\issue 1
\pages 27--65
\jour Math. USSR-Sb.
\yr 1971
\vol 13
\issue 1
\pages 25--64

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    This publication is cited in the following articles:
    1. M. S. Agranovich, “Theorem on matrices depending on parameters and its applications to hyperbolic systems”, Funct. Anal. Appl., 6:2 (1972), 85–93  mathnet  crossref  mathscinet  zmath
    2. B. R. Vainberg, “On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as $t\to\infty$ of solutions of non-stationary problems”, Russian Math. Surveys, 30:2 (1975), 1–58  mathnet  crossref  mathscinet  zmath
    3. A. G. Fedotov, “Ob ellipticheskikh zadachakh so spektralnym parametrom, polinomialno vkhodyaschim v granichnye usloviya”, UMN, 32:4(196) (1977), 269–270  mathnet  mathscinet  zmath
    4. L. R. Volevich, S. G. Gindikin, “The method of energy estimates in mixed problems”, Russian Math. Surveys, 35:5 (1980), 57–137  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. Nguyen Minh Chuong, “On isomorphism of Sobolev spaces of variable order”, Math. USSR-Sb., 49:1 (1984), 1–17  mathnet  crossref  mathscinet  zmath
    6. Roitberg YA., “Boundary-Value and Mixed Problems for General Hyperbolic Systems in a Complete Scale of Sobolev Type Spaces”, 318, no. 4, 1991, 820–824  isi
    7. S. Yu. Dobrokhotov, O. L. Tolstova, I. Yu. Chudinovich, “Waves in a fluid over an elastic bottom. The existence theorem and exact solutions”, Math. Notes, 54:6 (1993), 1208–1222  mathnet  crossref  mathscinet  zmath  isi
    8. B. A. Plamenevskii, “On the Wave Equation in a Cylinder with Edges”, Funct. Anal. Appl., 32:1 (1998), 63–65  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. B. A. Amosov, M. Sh. Birman, M. I. Vishik, L. R. Volevich, I. M. Gel'fand, L. F. Fridlender, M. A. Shubin, “Mikhail Semenovich Agranovich (on his 70th birthday)”, Russian Math. Surveys, 56:4 (2001), 777–784  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. S. I. Matyukevich, B. A. Plamenevskii, “Elastodynamics in domains with edges”, St. Petersburg Math. J., 18:3 (2007), 459–510  mathnet  crossref  mathscinet  zmath  elib
    11. V. B. Dmitriev, “A nonlocal problem for a first-order partial differential equation”, Russian Math. (Iz. VUZ), 56:4 (2012), 1–8  mathnet  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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