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Sobolev Gradients and Differential Equations

  • Book
  • © 1997

Overview

Authors:
  1. John William Neuberger
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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1670)

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Table of contents (18 chapters)

  1. Front Matter

    Pages I-VIII
    Download chapter PDF

  2. Several gradients

    • John William Neuberger
    Pages 1-3

  3. Comparison of two gradients

    • John William Neuberger
    Pages 5-9

  4. Continuous steepest descent in Hilbert space: Linear case

    • John William Neuberger
    Pages 11-13

  5. Continuous steepest descent in Hilbert space: Nonlinear case

    • John William Neuberger
    Pages 15-31

  6. Orthogonal projections, Adjoints and Laplacians

    • John William Neuberger
    Pages 33-42

  7. Introducing boundary conditions

    • John William Neuberger
    Pages 43-52

  8. Newton's method in the context of Sobolev gradients

    • John William Neuberger
    Pages 53-58

  9. Finite difference setting: the inner product case

    • John William Neuberger
    Pages 59-68

  10. Sobolev gradients for weak solutions: Function space case

    • John William Neuberger
    Pages 69-73

  11. Sobolev gradients in non-inner product spaces: Introduction

    • John William Neuberger
    Pages 75-78

  12. The superconductivity equations of Ginzburg-Landau

    • John William Neuberger
    Pages 79-91

  13. Minimal surfaces

    • John William Neuberger
    Pages 93-106

  14. Flow problems and non-inner product Sobolev spaces

    • John William Neuberger
    Pages 107-114

  15. Foliations as a guide to boundary conditions

    • John William Neuberger
    Pages 115-123

  16. Some related iterative methods for differential equations

    • John William Neuberger
    Pages 125-133

  17. A related analytic iteration method

    • John William Neuberger
    Pages 135-138

  18. Steepest descent for conservation equations

    • John William Neuberger
    Pages 139-140

  19. A sample computer code with notes

    • John William Neuberger
    Pages 141-143

  20. Back Matter

    Pages 145-150
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Keywords

About this book

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.

Bibliographic Information

  • Book Title: Sobolev Gradients and Differential Equations

  • Authors: John William Neuberger

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/BFb0092831

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1997

  • eBook ISBN: 978-3-540-69594-3Published: 13 November 2006

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: VIII, 152

  • Topics: Partial Differential Equations, Numerical Analysis

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