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Elliptic Mixed, Transmission and Singular Crack Problems
Gohar Harutyunyan, University of Oldenburg, Germany, and Bert-Wolfgang Schulze, University of Potsdam, Germany
A publication of the European Mathematical Society.
cover
EMS Tracts in Mathematics
2007; 777 pp; hardcover
Volume: 4
ISBN-10: 3-03719-040-X
ISBN-13: 978-3-03719-040-1
List Price: US$158
Member Price: US$126.40
Order Code: EMSTM/4
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Mixed, transmission, or crack problems belong to the analysis of boundary value problems on manifolds with singularities. The Zaremba problem with a jump between Dirichlet and Neumann conditions along an interface on the boundary is a classical example. The central theme of this book is to study mixed problems in standard Sobolev spaces as well as in weighted edge spaces where the interfaces are interpreted as edges. Parametrices and regularity of solutions are obtained within a systematic calculus of boundary value problems on manifolds with conical or edge singularities. This calculus allows singularities on the interface and homotopies between mixed and crack problems. Additional edge conditions are computed in terms of relative index results. In a detailed final chapter, the intuitive ideas of the approach are illustrated, and there is a discussion of future challenges. A special feature of the text is the inclusion of many worked-out examples which help the reader to appreciate the scope of the theory and to treat new cases of practical interest.

This book is addressed to mathematicians and physicists interested in models with singularities, associated boundary value problems, and their solvability strategies based on pseudo-differential operators. The material is also useful for students in higher semesters and young researchers, as well as for experienced specialists working in analysis on manifolds with geometric singularities, the applications of index theory and spectral theory, operator algebras with symbolic structures, quantisation, and asymptotic analysis.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in differential equations.

Table of Contents

  • Introduction
  • Boundary value problems with mixed and interface data
  • Symbolic structures and associated operators
  • Boundary value problems with the transmission property
  • Mixed problems in standard Sobolev spaces
  • Mixed problems in weighted edge spaces
  • Operators on manifolds with conical singularities and boundary
  • Operators on manifolds with edges and boundary
  • Corner operators and problems with singular interfaces
  • Operators in infinite cylinders and the relative index
  • Intuitive ideas of the calculus on singular manifolds
  • Bibliography
  • List of symbols
  • Index
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