Differential equations on singular manifolds. Semiclassical theory and operator algebras.

*(English)*Zbl 0946.58021
Mathematical Topics. 15. Weinheim: Wiley-VCH. 376 p. (1998).

The purpose of this book is to study linear partial differential equations on manifolds with a finite number of singular points of conical, cuspical, or more general type. Near a cusp, the asymptotic expansions of solutions are much more complicated than in the “classical” conical case. To treat these difficulties, the authors have to develop an appropriate framework.

The book is divided into 3 parts.

I) The first part of the book includes some general questions pertaining to the topics of the book on the whole. The different types of singularities dealt with are described. The connection between the asymptotic behavior of solutions at different singular points is studied.

II) In the second part of the book, the authors study elliptic equations on manifolds with singular points: the methods developed in the first part are applied; solutions are constructed and the Fredholm property is studied. Here, in particular, the construction of suitable operator algebras acting on suitable space scales is crucial; this construction is made with the help of Maslov’s non-commutative analysis.

III) The third part is devoted to the study of hyperbolic equations when the degeneracy of the “stationary” part of the operator may be conical as well as cuspidal. This part also contains a chapter on the vibration of elastic shells with conical points.

Finally, there are two appendices containing technical tools used in the book. The reader will find in this book much useful information on linear partial differential equations on singular manifolds.

The book is divided into 3 parts.

I) The first part of the book includes some general questions pertaining to the topics of the book on the whole. The different types of singularities dealt with are described. The connection between the asymptotic behavior of solutions at different singular points is studied.

II) In the second part of the book, the authors study elliptic equations on manifolds with singular points: the methods developed in the first part are applied; solutions are constructed and the Fredholm property is studied. Here, in particular, the construction of suitable operator algebras acting on suitable space scales is crucial; this construction is made with the help of Maslov’s non-commutative analysis.

III) The third part is devoted to the study of hyperbolic equations when the degeneracy of the “stationary” part of the operator may be conical as well as cuspidal. This part also contains a chapter on the vibration of elastic shells with conical points.

Finally, there are two appendices containing technical tools used in the book. The reader will find in this book much useful information on linear partial differential equations on singular manifolds.

Reviewer: P.Godin (Bruxelles)

##### MSC:

58J05 | Elliptic equations on manifolds, general theory |

58J45 | Hyperbolic equations on manifolds |

35B40 | Asymptotic behavior of solutions to PDEs |