A Sobolev gradient of a realvalued functional is a gradient of that functionaltaken relative to the underlying Sobolev norm. This book shows how descentmethods ... » 

A Sobolev gradient of a realvalued functional is a gradient of that functionaltaken relative to the underlying Sobolev norm. This book shows how descentmethods using such gradients allow a unified treatment of a wide variety ofproblems in differential equations. Equal emphasis is placed on numerical andtheoretical matters. Several concrete applications are made to illustrate themethod. These applications include 1 GinzburgLandau functionals ofsuperconductivity 2 problems of transonic flow in which type depends locallyon nonlinearities and 3 minimal surface problems. Sobolev gradientconstructions rely on a study of orthogonal projections onto graphs of closeddensely defined linear transformations from one Hilbert space to another. Thesedevelopments use work of Weyl von Neumann and Beurling. «