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＜E-Book＞
Sobolev Gradients and Differential Equations / by john neuberger
(Lecture Notes in Mathematics. ISSN:16179692 ; 1670)

Edition	1st ed. 1997.
Publisher	Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer
Year	1997
Language	English
Size	VIII, 152 p : online resource
Authors	*neuberger, john author SpringerLink (Online service)
Subjects	LCSH:Differential equations LCSH:Numerical analysis FREE:Differential Equations FREE:Numerical Analysis
Notes	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 gradients 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 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 HTTP:URL=https://doi.org/10.1007/BFb0092831

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