Differential operator equations. A method of model operators in the theory of boundary value problems. (Differentsial’no-operatornye uravneniya. Metod model’nykh operatorov v teorii granichnykh zadach.) (Russian. English summary) Zbl 0968.47012

Dezin, A. A.

Trudy Matematicheskogo Instituta Imeni V. A. Steklova. 229. Moskva: Nauka. Moskva: MAIK Nauka/Interperiodica. 175 p. (2000).

Author’s summary: “In this monograph a wide range of problems of the theory of linear partial differential equations are considered from a unified point of view. The procedure of reducing a problem to a model differential operator equation of a special simple structure is studied. Classical and nonclassical equations and problems are compared. The spectral characteristics and properties of generalized solutions are considered for mixed-type and degenerating equations as well as for equations with discontinuous coefficients and equations containing a small parameter. Considerable attention is paid to the questions of the general theory of boundary problems. Necessary information is given from functional analysis and spectral theory of operators. For specialists in mathematical physics, functional analysis and applied mathematics as well as for senior students and postgraduates of relevant specialties.”
As a general remark, the results are connected to those contained in the author’s monograph “Partial differential equations”, Springer-Verlag (1987; Zbl 0623.35005), but there are some differences of principle in the broaching of the notions. Also one can take notice on the fact that in the present monograph several times it is underlined the character of the punctual nature of the spectrum of the differential operators in the compact case which makes natural the use of Fourier series in contrast with another monographs using, as fundamental instrument of investigations, the Fourier transform and its generalization for pseudo-differential operators or the microlocal analysis.

Reviewer: Peetre Craciunas (Iaşi)

MSC:

47F05	General theory of partial differential operators
47E05	General theory of ordinary differential operators
34G10	Linear differential equations in abstract spaces
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
35A22	Transform methods (e.g., integral transforms) applied to PDEs
35G05	Linear higher-order PDEs
35G20	Nonlinear higher-order PDEs
35P05	General topics in linear spectral theory for PDEs
47N20	Applications of operator theory to differential and integral equations

Keywords:

model differential operator equation; generalized solutions for mixed-type and degeneratin equations; equations with discontinuous coefficients; small parameter; linear partial differential equations; spectral characteristics; general theory of boundary problems

Citations:

Zbl 0623.35005