Mathematics and Its Applications

Crack Theory and Edge Singularities

Authors: Kapanadze, D. V., Schulze, Bert-Wolfgang

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About this book

Boundary value problems for partial differential equations playa crucial role in many areas of physics and the applied sciences. Interesting phenomena are often connected with geometric singularities, for instance, in mechanics. Elliptic operators in corresponding models are then sin­ gular or degenerate in a typical way. The necessary structures for constructing solutions belong to a particularly beautiful and ambitious part of the analysis. Cracks in a medium are described by hypersurfaces with a boundary. Config­ urations of that kind belong to the category of spaces (manifolds) with geometric singularities, here with edges. In recent years the analysis on such (in general, stratified) spaces has become a mathematical structure theory with many deep relations with geometry, topology, and mathematical physics. Key words in this connection are operator algebras, index theory, quantisation, and asymptotic analysis. Motivated by Lame's system with two-sided boundary conditions on a crack we ask the structure of solutions in weighted edge Sobolov spaces and subspaces with discrete and continuous asymptotics. Answers are given for elliptic sys­ tems in general. We construct parametrices of corresponding edge boundary value problems and obtain elliptic regularity in the respective scales of weighted spaces. The original elliptic operators as well as their parametrices belong to a block matrix algebra of pseudo-differential edge problems with boundary and edge conditions, satisfying analogues of the Shapiro-Lopatinskij condition from standard boundary value problems. Operators are controlled by a hierarchy of principal symbols with interior, boundary, and edge components.

Table of contents (5 chapters)

  • Boundary value problems with the transmission property

    Kapanadze, David (et al.)

    Pages 1-82

  • Operators on manifolds with conical singularities

    Kapanadze, David (et al.)

    Pages 83-191

  • Operators on manifolds with exits to infinity

    Kapanadze, David (et al.)

    Pages 193-294

  • Boundary value problems on manifolds with edges

    Kapanadze, David (et al.)

    Pages 295-410

  • Crack theory

    Kapanadze, David (et al.)

    Pages 411-454

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-94-017-0323-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-1-4020-1524-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for USA
  • ISBN 978-90-481-6384-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Crack Theory and Edge Singularities
Authors
Series Title
Mathematics and Its Applications
Series Volume
561
Copyright
2003
Publisher
Springer Netherlands
Copyright Holder
Springer Science+Business Media Dordrecht
eBook ISBN
978-94-017-0323-9
DOI
10.1007/978-94-017-0323-9
Hardcover ISBN
978-1-4020-1524-3
Softcover ISBN
978-90-481-6384-7
Edition Number
1
Number of Pages
XXVII, 485
Topics