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Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems

Michael Struwe
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Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 34
[Reviewed by
John D. Cook
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The fourth edition of Michael Struwe’s book Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems was published in 2008, 18 years after the first edition. It contains a great deal of information in under 300 pages, including recent results in such areas as the Yamabe problem. The bibliography alone would make it a valuable reference as it contains nearly 500 references. Entries range from a seminal 1744 paper by Leonard Euler to a paper by Struwe that was in press at the same time as his book.

The variational methods presented in Struwe’s book have their roots in the calculus of variations initiated by Leonard Euler in the eighteenth century. Whereas elementary calculus studies functions of a real variable, the calculus of variations studies operators that act on functions. So while elementary calculus may seek to find numbers that minimize a function of one variable, calculus of variations seeks to find functions that minimize an operator subject to constraints. These operators often have physical or geometric significance. For example, the goal may be to find a path that requires minimal energy or the surface of minimum area for a specified boundary.

Variational methods have become very sophisticated since they were first introduced two and a half centuries ago. A reader for whom functional analysis and Sobolev spaces are not second nature will not find this book accessible. The modern approach to studying partial differential equations begins the search for solutions in spaces of functions (Sobolev spaces) that may satisfy the equation in a generalized sense. A common technique is to use results from functional analysis to prove that a sequence of approximate solutions converge to a candidate solution in a weak sense. Then hard analysis is used to show that the weak limit satisfies the differential equation in a stronger sense. The process is analogous to finding a rational solution to an equation by first extending the problem to real numbers, using calculus techniques to find a real solution, then proving that the real solution must in fact be rational. Sobolev spaces complete spaces of smooth functions in a manner analogous to the way real numbers complete the rationals. And functional analysis provides powerful tools for working in these function spaces, analogous to the way calculus provides tools for working with real functions.

Struwe’s book is addressed to researchers in differential geometry and partial differential equations. There is a small amount of expository material, for example explaining the connection between soap bubbles and variational methods, but Struwe hits the ground running from the very beginning. For example, a sentence on the first page of the first chapter begins “Suppose E is a Fréchet differentiable functional on a Banach space with normed dual V*…” The book starts with well-established results for elliptic partial differential equations on open subsets of Rn and concludes with problems in areas of active research posed on Riemannian manifolds.

John D. Cook began his career working in partial differential equations and now works in biostatistics at M. D. Anderson Cancer Center.

The Direct Methods in the Calculus of Variations.- Lower Semi-Continuity.- Constraints.- Compensated Compactness.- The Concentration-Compactness Principle.- Ekeland's Variational Principle.- Duality.- Minimization Problems Depending on Parameters.- Minimax Methods.- The Finite Dimensional Case.- The Palais-Smale Condition.- A General Deformation Lemma.- The Minimax Principle.- Index Theory.- The Mountain Pass Lemma and its Variants.- Perturbation Theory.- Linking.- Parameter Dependence.- Critical Points of Mountain Pass Type.- Non-Differentiable Functionals.- Ljusternik-Schnirelman Theory on Convex Sets.- Limit Cases of the Palais-Smale Condition.- Pohozaev's Non-Existence Result.- The Brezis-Nierenberg Result.- The Effect of Topology.- The Yamabe Problem.- The Dirichlet Problem for the Equation of Constant Mean Curvature.- Harmonic Maps of Riemannian Surfaces.- Appendix A.- Appendix B.- Appendix C.- References.- Index.