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Sobolev Gradients and Differential Equations

  • Authors
  • John¬†William¬†Neuberger

Part of the Lecture Notes in Mathematics book series (LNM, volume 1670)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. John William Neuberger
    Pages 1-3
  3. John William Neuberger
    Pages 5-9
  4. John William Neuberger
    Pages 33-42
  5. John William Neuberger
    Pages 43-52
  6. John William Neuberger
    Pages 53-58
  7. John William Neuberger
    Pages 59-68
  8. John William Neuberger
    Pages 79-91
  9. John William Neuberger
    Pages 93-106
  10. John William Neuberger
    Pages 107-114
  11. John William Neuberger
    Pages 115-123
  12. John William Neuberger
    Pages 125-133
  13. John William Neuberger
    Pages 135-138
  14. John William Neuberger
    Pages 139-140
  15. John William Neuberger
    Pages 141-143
  16. Back Matter
    Pages 145-150

About this book

Introduction

A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling.

Keywords

Newton's method Sobolev space differential equation numerical analysis orthogonal projections partial differential equation partial differential equations sobolev gradient

Bibliographic information

  • DOI https://doi.org/10.1007/BFb0092831
  • Copyright Information Springer-Verlag Berlin Heidelberg 1997
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-63537-6
  • Online ISBN 978-3-540-69594-3
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site