Sobolev gradient and differential equations
By: J.W. Neuberger.
Series: Lecture notes in mathematics -- 1670
Publisher: Springer
Published: 1997
Place of publication: Berlin; London
Language: eng
Number of pages: 149
Subjects: Sobolev gradients.; Differential equations -- Numerical solutions.
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.
Notes: Includes bibliographical references and index.