Sobolev Gradients and Differential Equations
Published on 10. October 1997
Book
Paperback/Softback
VIII, 152 pages
978-3-540-63537-6 (ISBN)
Description
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.
More details
Series
Language
English
Place of publication
Heidelberg
Germany
Publishing group
Springer Berlin
Target group
College/higher education
Professional and scholarly
Product notice
Paperback (trade)
Illustrations
8 figures
Dimensions
Height: 23.4 cm
Width: 15.6 cm
Thickness: 8 mm
Weight
236 gr
ISBN-13
978-3-540-63537-6 (9783540635376)
DOI
10.1007/BFb0092831
Schweitzer Classification
Other editions
New editions
john neuberger
Sobolev Gradients and Differential Equations
Book
12/2009
2nd Edition
Springer
€53.49
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Content
Several Gradients.- Comparison of Two Gradients.- Continuous Steepest Descent in Hilbert Space: Linear Case.- Continuous Steepest Descent in Hilbert Space: Nonlinear Case.- Orthogonal Projections, Adjoints and Laplacians.- Introducing Boundary Conditions.- Newton's Method in the Context of Sobolev Gradients.- Finite Difference Setting: the Inner Product Case.- Sobolev Gradients for Weak Solutions: Function Space Case.- Sobolev Gradient in Non-inner Product Spaces: Introduction.- The Superconductivity Equations of Ginzburg-Landau.- Minimal Surfaces.- Flow Problems and Non-inner Product Sobolev Spaces.- Foliations as a Guide to Boundary Conditions.- Some Related Iterative Methods for Differential Equations.- A Related Analytic Iteration Method.- Steepest Descent for Conservation Equations.- A Sample Computer Code with Notes.- Bibliography.- Index.