Sobolev Gradients and Differential Equations

John William Neuberger (auth.)3540635378, 9783540635376

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.

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Table of contents :
Several gradients….Pages 1-3
Comparison of two gradients….Pages 5-9
Continuous steepest descent in Hilbert space: Linear case….Pages 11-13
Continuous steepest descent in Hilbert space: Nonlinear case….Pages 15-31
Orthogonal projections, Adjoints and Laplacians….Pages 33-42
Introducing boundary conditions….Pages 43-52
Newton’s method in the context of Sobolev gradients….Pages 53-58
Finite difference setting: the inner product case….Pages 59-68
Sobolev gradients for weak solutions: Function space case….Pages 69-73
Sobolev gradients in non-inner product spaces: Introduction….Pages 75-78
The superconductivity equations of Ginzburg-Landau….Pages 79-91
Minimal surfaces….Pages 93-106
Flow problems and non-inner product Sobolev spaces….Pages 107-114
Foliations as a guide to boundary conditions….Pages 115-123
Some related iterative methods for differential equations….Pages 125-133
A related analytic iteration method….Pages 135-138
Steepest descent for conservation equations….Pages 139-140
A sample computer code with notes….Pages 141-143