Progress In Variational Methods In Hamiltonian Systems And Elliptic Equations

This research note gives a comprehensive account of the use of variational methods in the study of Hamiltonian systems and elliptic equations.

Hilberts talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateaus problem by Douglas and Rad. This third edition gives a concise introduction to variational methods and presents an overview of areas of current research in the field, plus a survey on new developments.

In the framework of the "Annee non lineaire" (the special nonlinear year) sponsored by the C.N.R.S. (the French National Center for Scien tific Research), a meeting was held in Paris in June 1988. It took place in the Conference Hall of the Ministere de la Recherche and had as an organizing theme the topic of "Variational Problems." Nonlinear analysis has been one of the leading themes in mathemat ical research for the past decade. The use of direct variational methods has been particularly successful in understanding problems arising from physics and geometry. The growth of nonlinear analysis is largely due to the wealth of ap plications from various domains of sciences and industrial applica tions. Most of the papers gathered in this volume have their origin in applications: from mechanics, the study of Hamiltonian systems, from physics, from the recent mathematical theory of liquid crystals, from geometry, relativity, etc. Clearly, no single volume could pretend to cover the whole scope of nonlinear variational problems. We have chosen to concentrate on three main aspects of these problems, organizing them roughly around the following topics: 1. Variational methods in partial differential equations in mathemat ical physics 2. Variational problems in geometry 3. Hamiltonian systems and related topics.

In the last forty years, nonlinear analysis has been broadly and rapidly developed. Lectures presented in the International Conference on Variational Methods at the Chern Institute of Mathematics in Tianjin of May 2009 reflect this development from different angles. This volume contains articles based on lectures in the following areas of nonlinear analysis: critical point theory, Hamiltonian dynamics, partial differential equations and systems, KAM theory, bifurcation theory, symplectic geometry, geometrical analysis, and celestial mechanics. Combinations of topological, analytical (especially variational), geometrical, and algebraic methods in these researches play important roles. In this proceedings, introductory materials on new theories and surveys on traditional topics are also given. Further perspectives and open problems on hopeful research topics in related areas are described and proposed. Researchers, graduate and postgraduate students from a wide range of areas in mathematics and physics will find contents in this proceedings are helpful.

A collection of up-to-date research and classic papers reflecting the work of Michael Herman.

The articles in this volume are an outgrowth of an international conference entitled Variational and Topological Methods in the Study of Nonlinear Phe- nomena, held in Pisa in January-February 2000. Under the framework of the research project Differential Equations and the Calculus of Variations, the conference was organized to celebrate the 60th birthday of Antonio Marino, one of the leaders of the research group and a significant contrib- utor to the mathematical activity in this area of nonlinear analysis. The volume highlights recent advances in the field of nonlinear functional analysis and its applications to nonlinear partial and ordinary differential equations, with particular emphasis on variational and topological meth- ods. A broad range of topics is covered, including: concentration phenomena in PDEs, variational methods with applications to PDEs and physics, pe- riodic solutions of ODEs, computational aspects in topological methods, and mathematical models in biology. Though well-differentiated, the topics covered are unified through a com- mon perspective and approach. Unique to the work are several chapters on computational aspects and applications to biology, not usually found with such basic studies on PDEs and ODEs. The volume is an excellent reference text for researchers and graduate students in the above mentioned fields. Contributors are M. Clapp, M.J. Esteban, P. Felmer, A. Ioffe, W. Marzan- towicz, M. Mrozek, M. Musso, R. Ortega, P. Pilarczyk, M. del Pino, E. Sere, E. Schwartzman, P. Sintzoff, R. Turner, and I\f. Willem.