Title: Sobolev gradient and differential equations
First published: 1997
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.
Subjects: Differential equations, Numerical solutions, Sobolev gradients, Gradients de Sobolev, Solutions numériques, Équations différentielles, Gradientenverfahren, Partiële differentiaalvergelijkingen, Partielle Differentialgleichung, Mathematics, Differential equations, partial, Numerical analysis
LC classification: QA3, QA372