An Invitation to Variational Methods in Differential Equations (Birkhauser Advanced Texts / Basler Lehrbucher)

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9780817645359: An Invitation to Variational Methods in Differential Equations (Birkhauser Advanced Texts / Basler Lehrbucher)
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This textbook introduces variational methods and their applications to differential equations to graduate students and researchers interested in differential equations and nonlinear analysis. It serves as a sampling of topics in critical point theory. Coverage includes: minimizations, deformations results, the mountain-pass and saddle-point theorems, critical points under constraints, and issues of compactness. Applications immediately follow each result for easy assimilation by the reader. This straightforward and systematic presentation includes many exercises and examples to motivate the study of variational methods.

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This book is a short introductory text to variational techniques with applications to differential equations. It presents a sampling of topics in critical point theory with applications to existence and multiplicity of solutions in nonlinear problems involving ordinary differential equations (ODEs) and partial differential equations (PDEs).

Five simple problems in ODEs which illustrate existence of solutions from a variational point of view are introduced in the first chapter. These problems set the stage for the topics covered, including minimization, deformation results, the mountain-pass theorem, the saddle-point theorem, critical points under constraints, a duality principle, critical points in the presence of symmetry, and problems with lack of compactness. Each topic is presented in a straightforward manner, and followed by one or two illustrative applications.

The concise, straightforward, user-friendly approach of this textbook will appeal to graduate students and researchers interested in differential equations, analysis, and functional analysis.

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"This book is intended to be an introduction to variational methods for ODEs and PDEs. ... Overall, this is a well-written book on the variational methods ... . The author clearly loves and knows the subject area and it is very good at providing an overview to the area ... . As such, a good guided reading course for graduate students could be made from this book, covering one chapter per session (or two)." (David A. W. Barton, Dynamical Systems Magazine, April, 2008)

"This little book consists of a very clear introduction to variational methods and their applications to ODEs and PDEs. ... the bibliography contains basic items on the subject and also turns out to be very useful for further reading. In my opinion the book should be strongly recommended to anyone―graduate student or researcher―who is interested in variational methods and their applications to differential equations." (Salvatore A. Marano, Mathematical Reviews, Issue 2008 k)

“The intention of the author is to provide a first introduction to variational methods in solving ODEs and PDEs for graduate students. The book is a concise collection of some fundamental aspects of this area ... . It presents a nice and direct approach to these central topics and is written with great care and clarity. It can be warmly recommended to anyone wishing to enter this active area of research.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)

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Publisher: Springer Verlag Gmbh, 2010
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