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Uspekhi Mat. Nauk, 1969, Volume 24, Issue 1(145), Pages 61–125 (Mi umn5452)  

This article is cited in 9 scientific papers (total in 10 papers)

Boundary value problrms for systems of first order pseudodifferential operators

M. S. Agranovich

Abstract: A large group of problems for systems of partial differential equations of the first order is studied by common methods; for these problems the theorem on energy inequalities is proved, under the assumption that the system (or rather, its characteristic matrix) is symmetric, and also the theorem on the identity of the weak and the strong solutions; these two theorems are used to prove existence and uniqueness of the strong solution. The methods are applicable to a number of problems for symmetric hyperbolic systems of the first order and for symmetric stationary systems that need not be elliptic. Recently new possibilities of developing and applying these methods by using pseudodifferential operators have been discovered, and these are far from being exhausted at the present time. In § 1 the problem is stated and a brief survey of the literature is given. In §§ 2–6 the three theorems mentioned above are set out with proofs suitable for systems of pseudodifferential operators of the first order in a bounded domain. § 7 deals with boundary value problems for symmetrizable systems, more general than the symmetric systems.

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English version:
Russian Mathematical Surveys, 1969, 24:1, 59–126

Bibliographic databases:

UDC: 517.9+517.4
MSC: 35S15, 35F15
Received: 20.06.1968

Citation: M. S. Agranovich, “Boundary value problrms for systems of first order pseudodifferential operators”, Uspekhi Mat. Nauk, 24:1(145) (1969), 61–125; Russian Math. Surveys, 24:1 (1969), 59–126

Citation in format AMSBIB
\by M.~S.~Agranovich
\paper Boundary value problrms for systems of first order pseudodifferential operators
\jour Uspekhi Mat. Nauk
\yr 1969
\vol 24
\issue 1(145)
\pages 61--125
\jour Russian Math. Surveys
\yr 1969
\vol 24
\issue 1
\pages 59--126

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    This publication is cited in the following articles:
    1. M. S. Agranovich, “Boundary value problems for systems with a parameter”, Math. USSR-Sb., 13:1 (1971), 25–64  mathnet  crossref  mathscinet  zmath
    2. A. V. Babin, “A formula expressing the solution of a differential equation with analytic coefficients on a manifold without boundary in terms of the data of the problem”, Math. USSR-Sb., 30:4 (1976), 539–563  mathnet  crossref  mathscinet  zmath  isi
    3. A. V. Babin, “An expression for the solution of a differential equation in terms of iterates of differential operators”, Math. USSR-Sb., 34:4 (1978), 411–424  mathnet  crossref  mathscinet  zmath
    4. Daniel Gourdin, Rue J. Jaurès, “Probleme de cauchy non caracteristique pour les systemes hyperboliques a caracteristiques de multiplicite variable domaine de dependance”, Communications in Partial Differential Equations, 4:5 (1979), 447  crossref
    5. 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
    6. I. Lasiecka, R. Triggiani, “A cosine operator approach to modelingL 2(0,T; L 2 (Γ))—Boundary input hyperbolic equations”, Appl Math Optim, 7:1 (1981), 35  crossref  mathscinet  zmath  isi
    7. A. A. Pankov, “On bounded and almost-periodic solutions of certain non-linear evolution equations”, Russian Math. Surveys, 37:2 (1982), 239–240  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. A. A. Pankov, “Bounded solutions, almost periodic in time, of a class of nonlinear evolution equations”, Math. USSR-Sb., 49:1 (1984), 73–86  mathnet  crossref  mathscinet  zmath
    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. E. V. Radkevich, “On well-posedness of the Cauchy Problem and the Mixed Problem for some class of hyperbolic systems and equations with constant coefficients and variable multiplicity of characteristics”, Journal of Mathematical Sciences, 149:5 (2008), 1580–1607  mathnet  crossref  mathscinet  zmath
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