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Topological tools in Nonlinear Analysis had a tremendous develop ment during the last few decades. The three main streams of research in this field, Topological Degree, Singularity Theory and Variational Meth ods, have lately become impetuous rivers of scientific investigation. The process is still going on and the achievements in this area are spectacular. A most promising and rapidly developing field of research is the study of the role that symmetries play in nonlinear problems. Symmetries appear in a quite natural way in many problems in physics and in differential or symplectic geometry, such as closed orbits for autonomous Hamiltonian systems, configurations of symmetric elastic plates under pressure, Hopf Bifurcation, Taylor vortices, convective motions of fluids, oscillations of chemical reactions, etc . . . Some of these problems have been tackled recently by different techniques using equivariant versions of Degree, Singularity and Variations. The main purpose of the present volume is to give a survey of some of the most significant achievements obtained by topological methods in Nonlinear Analysis during the last two-three decades. The survey articles presented here reflect the personal taste and points of view of the authors (all of them well-known and distinguished specialists in their own fields) on the subject matter. A common feature of these papers is that of start ing with an historical introductory background of the different disciplines under consideration and climbing up to the heights of the most recent re sults.
Contents
Variational Methods and Nonlinear Problems: Classical Results and Recent Advances.- • Introduction.- • Lusternik-Schnirelman Theory.- • Applications to Nonlinear Eigenvalues.- • Unbounded Functionals.- • Elliptic Dirichlet Problems.- • Singular Potentials.- • References.- to Morse Theory: A New Approach.- • Introduction.- • Contents.- • The Abstract Theory.- • The Morse Index.- • The Poincaré Polynomial.- • The Conley Blocks.- • The Morse Relations.- • Morse Theory for Degenerate Critical Points.- • Some Existence Theorems.- • An Application to Riemannian Geometry.- • Riemannian Manifolds.- • Geodesies.- • The Morse Theory for Geodesics.- • The Index Theorem.- • An Application to Space-Time Geometry.- • Introduction.- • Some Examples of Lorentzian Manifolds.- • Morse Theory for Lorentzian Manifolds.- • Preliminary Lemmas.- • Proof of The Morse Relations For Static Space-Time.- • Some Application to a Semilinear Elliptic Equation.- • Introduction.- • The Sublinear Case.- • The Superlinear Case Morse Relations for Positive Solutions.- • The Functional Setting.- • Some Hard Analysis.- • The Photography Method.- • The Topology of The Strip.- • References.- Applications of Singularity Theory to the Solutions of Nonlinear Equations.- • The Full Lyapunov-Schmidt Reduction.- • Mather's Theory of C?-Stability of Mappings - Global Theory.- • Mather's Local Theory as Paradigm.- • Singularity Theory with Special Conditions.- • The Structure of Nonlinear Fredholm Operators.- • Multiplicities of Solutions to Nonlinear Equations.- • The Theory for Topological Equivalence.- • Bibliography.- Fixed Point Index Calculations and Applications.- • The Fixed Point Index.- • Some Remarks onConvex Sets.- • A Basic Index Calculation.- • Index Calculations in Product Cones.- • Applications of Index Formulae - I.- • Applications of Index Formulae - II.- • Some Global Branches.- • Monotone Dynamical Systems.- • Preliminaries.- • Connecting Orbits and Related Results.- • Generic Convergence.- • References.- Topological Bifurcation.- • Abstract.- • Introduction.- • Preliminaries.- • One Parameter Bifurcation.- • Local Bifurcation.- • Global Bifurcation.- • Special Nonlinearities.- • Multiparameter Bifurcation.- • Sufficient Conditions for Local Bifurcation.- • Necessary Conditions for Linearized Local Bifurcation.- • Multiparameter Global Bifurcation.- • A Summation Formula and A Generalized Degree.- • Structure and Dimension of Global Branches.- • O-EPI Maps.- • Dimension.- • Application to Bifurcation Problems.- • Equivariant Bifurcation.- • Preliminaries.- • Consequences of the Symmetry.- • ?-EPI Maps.- • ?-Degree.- • The Equivariant J-Homomorphism and Sufficient Conditions.- • Necessary and Sufficient Conditions for Equivariant Bifurcation.- • Bibliography.- Critical Point Theory.- • Introduction.- • The Mountain Pass Theorem.- • The Saddle Point Theorem.- • Linking and A General Critical Point Theorem.- • Periodic Solutions of Hamiltonian Systems.- • Introduction.- • The Technical Framework.- • Periodic Solutions of Prescribed Energy.- • Periodic Solutions of Prescribed Period.- • Connecting Orbits.- • Introduction.- • Homoclinic Solutions.- • Heteroclinic Solutions.- • References.- Symplectic Topology: An Introduction.- • The Classical Uncertainty Principle, Symplectic Rigidity.- • Construction of Symplectic Invariants.- • Generating Functions.- •Historical Remarks.- • Appendix: Rigidity for Finite Dimensional Lie Groups.
