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Pseudo-differential boundary value problems, conical singularities and asymptotics. (English) Zbl 0810.35175

Mathematical Topics. 4. Berlin: Akademie Verlag. 580 p. (1994).
This book contains the exposition of part of B.-W. Schulze’s work on the calculus of pseudo-differential operators on manifolds with singularities. More precisely, the book is devoted to the case of conical singularities and the study of boundary value problems. Related ideas are developed in [B.-W. Schulze, Pseudo-differential operators on manifolds with singularities (1991; Zbl 0747.58003)] by the same author, and the reader is sometimes referred to this book for proofs.
The first part of the book is entitled “Mellin pseudo-differential operators”. There the author introduces and studies Sobolev-type spaces and pseudo-differential operators defined by means of the classical Mellin transform. Manifolds with conical singularities are introduced; ellipticity and parametrices of the pseudo-differential operators are discussed.
In the second part of the book, entitled “Pseudo-differential boundary value problems”, the author first shows how to express usual pseudo- differential operators on a half axis in terms of his Mellin-type pseudo- differential operators. Then a boundary symbolic calculus (using Melling transform) is constructed. Attention is paid to the transmission property which was introduced in the study of pseudo-differential boundary value problems in the sixties.

MSC:

35S15 Boundary value problems for PDEs with pseudodifferential operators
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations

Citations:

Zbl 0747.58003
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