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

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

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
\Bibitem{Agr69} \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 \mathnet{http://mi.mathnet.ru/umn5452} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=611008} \zmath{https://zbmath.org/?q=an:0175.10804|0193.06605} \transl \jour Russian Math. Surveys \yr 1969 \vol 24 \issue 1 \pages 59--126 \crossref{https://doi.org/10.1070/RM1969v024n01ABEH001340} 

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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. M. S. Agranovich, “Boundary value problems for systems with a parameter”, Math. USSR-Sb., 13:1 (1971), 25–64
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
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
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
5. L. R. Volevich, S. G. Gindikin, “The method of energy estimates in mixed problems”, Russian Math. Surveys, 35:5 (1980), 57–137
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
7. A. A. Pankov, “On bounded and almost-periodic solutions of certain non-linear evolution equations”, Russian Math. Surveys, 37:2 (1982), 239–240
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
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
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
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