Boundary value problems and singular pseudo-differential operators.

*(English)*Zbl 0907.35146
Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts. Chichester: John Wiley & Sons (ISBN 0-471-97557-5/hbk). xvi, 637 p. (1998).

The book is devoted to partial differential operators, and pseudo-differential operators, on manifolds with singularities. It can be regarded as natural continuation of two preceding books of the author [Pseudo-differential operators on manifolds with singularities (North-Holland, Amsterdam, 1991; Zbl 0747.58003), Pseudo-differential boundary value problems, conical singularities, and asymptotics (Akademie Verlag, Berlin 1994; Zbl 0810.35175)], with relation also to the text of Yu. V. Egorov and B.-W. Schulze [Pseudo-differential operators, singularities, applications (BirkhĂ¤user Verlag, Basel 1997; Zbl 0877.35141)].

With respect to these contributions, here many improvements and new details are given, which make the theory more accessible. Moreover, addressing here to a large audience, the author presents the material in a self-contained style, with examples and remarks. In particular, in Chapter 1, Section 1, the classical pseudo-differential calculus is reviewed, including all relevant proofs. The other Sections in Chapter 1 are devoted to special topics needed in the sequel of the book: parameter dependent pseudo-differential operators, pseudo-differential operators with vector valued symbols and with cone-shaped exists to infinity.

The core of the book is in the next Chapters 2 and 3, where the author treats manifolds with conical singularities and, respectively, with edges. We recall that, in the case of conical singularities, the basic local model is given by \[ A=t^{-m}\sum^m_{j=0} a_j(t)(-tD_t)^j,\quad t\in \mathbb{R}_+, \] where \(a_j(t)\) are partial differential operators of order \(m-j\) on a smooth manifold \(X\). In the case of a manifold with edges, the local model is \[ A=t^{-m}\sum_{j+|\alpha|\leq m} a_{j\alpha} (t,y)(-tD_t)^j(tD_y)^\alpha \] where \(t\in\mathbb{R}_+\), \(y\in\Omega\subset\mathbb{R}^q\) and \(a_{j\alpha}(t,y)\) are partial differential operators of order \(m-j-|\alpha|\) on the smooth manifold \(X\).

The general theory is presented in detail, including definition of weighted Sobolev spaces, symbolic calculus, parametric construction in the pseudo-differential frame.

Chapter 4, finally, concerns pseudo-differential boundary value problems with the transmission property, obtained as a modification of the wedge theory.

The whole material is well-organized and the exposition is clear. The book can be recommended to researchers entering the field of the singular pseudo-differential operators, as well as to practitioners looking for a general information on the subject.

With respect to these contributions, here many improvements and new details are given, which make the theory more accessible. Moreover, addressing here to a large audience, the author presents the material in a self-contained style, with examples and remarks. In particular, in Chapter 1, Section 1, the classical pseudo-differential calculus is reviewed, including all relevant proofs. The other Sections in Chapter 1 are devoted to special topics needed in the sequel of the book: parameter dependent pseudo-differential operators, pseudo-differential operators with vector valued symbols and with cone-shaped exists to infinity.

The core of the book is in the next Chapters 2 and 3, where the author treats manifolds with conical singularities and, respectively, with edges. We recall that, in the case of conical singularities, the basic local model is given by \[ A=t^{-m}\sum^m_{j=0} a_j(t)(-tD_t)^j,\quad t\in \mathbb{R}_+, \] where \(a_j(t)\) are partial differential operators of order \(m-j\) on a smooth manifold \(X\). In the case of a manifold with edges, the local model is \[ A=t^{-m}\sum_{j+|\alpha|\leq m} a_{j\alpha} (t,y)(-tD_t)^j(tD_y)^\alpha \] where \(t\in\mathbb{R}_+\), \(y\in\Omega\subset\mathbb{R}^q\) and \(a_{j\alpha}(t,y)\) are partial differential operators of order \(m-j-|\alpha|\) on the smooth manifold \(X\).

The general theory is presented in detail, including definition of weighted Sobolev spaces, symbolic calculus, parametric construction in the pseudo-differential frame.

Chapter 4, finally, concerns pseudo-differential boundary value problems with the transmission property, obtained as a modification of the wedge theory.

The whole material is well-organized and the exposition is clear. The book can be recommended to researchers entering the field of the singular pseudo-differential operators, as well as to practitioners looking for a general information on the subject.

Reviewer: L.Rodino (Torino)

##### MSC:

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

35S15 | Boundary value problems for PDEs with pseudodifferential operators |

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

47G30 | Pseudodifferential operators |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |