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**Asymptotic methods in equations of mathematical physics. Transl. from the Russian by E. Primrose.**
*(English)*
Zbl 0743.35001

New York etc.: Gordon and Breach Science Publishers. vii, 498 p. (1989).

[For the Russian original 1982 edition see Zbl 0518.35002.]

The book under review is devoted to a systematic study of some asymptotic methods used in the analysis of linear stationary and nonstationary PDEs in mathematical physics. In the first part of the book, i.e. Chapters I–V, the author presents with great care and in detail several problems, methods and results in the field which became classical due to their applications. More precisely, in Chapter I is described and justified the stationary phase method, while Chapter II contains the simplest version of the WKB method for ODEs. Chapter III is devoted to a method of solving first order ODEs as for instance the Hamilton Jacobi equations. Chapter IV is concerned with the propagation of singularities of the Cauchy problem, as well as with the construction of corresponding formal asymptotic solutions for rapidly oscillating initial data. In Chapter V Maslov’s canonical operator method is presented via some simple examples, and a description of a method of constructing such canonical operators is included.

The second part of the book, i.e. Chapters VI–XII is more specialized and is mainly based on some of the author’s own contributions in the field. The most important results in the theory of elliptic equations in a bounded domain are included in Chapter VI, while in Chapter VII the same problems are analyzed for elliptic and hypoelliptic systems with constant coefficients in \(\mathbb R^ n\). Here some conditions at infinity of the type of radiation conditions are assumed in order to get existence and uniqueness. In Chapter VIII all these results are reformulated in order to handle exterior problems for elliptic and hypoelliptic systems with rapidly stabilizing coefficients at infinity.

Chapters IX and X are devoted to exterior elliptic problems obtained from a mixed problem for a hyperbolic system. Here the operator \(i\partial/\partial t\) is replaced by the spectral parameter \(k\) and the analyticity of solutions with respect to \(k\) is investigated. In Chapter XI the quasi-classical asymptotic behaviour of the solution to the problem of scattering of plane waves in inhomogeneous media and the quasi-classical asymptotic behaviour of the amplitude scattering are studied. In Chapter XII a new formula for the global parametrix of the Cauchy problem for hyperbolic systems in \(\mathbb R^ n\) is established. Using this formula a complete asymptotic expansion for \(\lambda\to\infty\) of the spectral function is obtained: (1) in the case of a second order self-adjoint elliptic operator; (2) in the case of a system of first order equations in \(\mathbb R^ n\).

The book under review is devoted to a systematic study of some asymptotic methods used in the analysis of linear stationary and nonstationary PDEs in mathematical physics. In the first part of the book, i.e. Chapters I–V, the author presents with great care and in detail several problems, methods and results in the field which became classical due to their applications. More precisely, in Chapter I is described and justified the stationary phase method, while Chapter II contains the simplest version of the WKB method for ODEs. Chapter III is devoted to a method of solving first order ODEs as for instance the Hamilton Jacobi equations. Chapter IV is concerned with the propagation of singularities of the Cauchy problem, as well as with the construction of corresponding formal asymptotic solutions for rapidly oscillating initial data. In Chapter V Maslov’s canonical operator method is presented via some simple examples, and a description of a method of constructing such canonical operators is included.

The second part of the book, i.e. Chapters VI–XII is more specialized and is mainly based on some of the author’s own contributions in the field. The most important results in the theory of elliptic equations in a bounded domain are included in Chapter VI, while in Chapter VII the same problems are analyzed for elliptic and hypoelliptic systems with constant coefficients in \(\mathbb R^ n\). Here some conditions at infinity of the type of radiation conditions are assumed in order to get existence and uniqueness. In Chapter VIII all these results are reformulated in order to handle exterior problems for elliptic and hypoelliptic systems with rapidly stabilizing coefficients at infinity.

Chapters IX and X are devoted to exterior elliptic problems obtained from a mixed problem for a hyperbolic system. Here the operator \(i\partial/\partial t\) is replaced by the spectral parameter \(k\) and the analyticity of solutions with respect to \(k\) is investigated. In Chapter XI the quasi-classical asymptotic behaviour of the solution to the problem of scattering of plane waves in inhomogeneous media and the quasi-classical asymptotic behaviour of the amplitude scattering are studied. In Chapter XII a new formula for the global parametrix of the Cauchy problem for hyperbolic systems in \(\mathbb R^ n\) is established. Using this formula a complete asymptotic expansion for \(\lambda\to\infty\) of the spectral function is obtained: (1) in the case of a second order self-adjoint elliptic operator; (2) in the case of a system of first order equations in \(\mathbb R^ n\).

Reviewer: I. Vrabie (Iaşi)

### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35C20 | Asymptotic expansions of solutions to PDEs |

81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |

70H20 | Hamilton-Jacobi equations in mechanics |

35J25 | Boundary value problems for second-order elliptic equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

65H10 | Numerical computation of solutions to systems of equations |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

35L45 | Initial value problems for first-order hyperbolic systems |

34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |