Topological nonlinear analysis : degree, singularity, and variations

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Topological nonlinear analysis : degree, singularity, and variations

Michele Matzeu, Alfonso Vignoli, editors

（Progress in nonlinear differential equations and their applications / editor, Haim Brezis, v. 15, 27）

Birkhäuser, c1995-c1997

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Includes bibliographical references

Description and Table of Contents

Volume

[1] : us ISBN 9780817637422

Description

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.

Table of 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.
Volume

2 : us ISBN 9780817638863

Description

The main purpose of the present volume is to give a survey of some of the most significant achievements obtained by topological methods in nonlin ear analysis during the last three decades. It is intended, at least partly, as a continuation of Topological Nonlinear Analysis: Degree, Singularity and Varia tions, published in 1995. The survey articles presented are concerned with three main streams of research, that is topological degree, singularity theory and variational methods, They reflect the personal taste of the authors, all of them well known and distinguished specialists. A common feature of these articles is to start with a historical introduction and conclude with recent results, giving a dynamic picture of the state of the art on these topics. Let us mention the fact that most of the materials in this book were pre sented by the authors at the "Second Topological Analysis Workshop on Degree, Singularity and Variations: Developments of the Last 25 Years," held in June 1995 at Villa Tuscolana, Frascati, near Rome. Michele Matzeu Alfonso Vignoli Editors Topological Nonlinear Analysis II Degree, Singularity and Variations Classical Solutions for a Perturbed N-Body System Gianfausto Dell 'A ntonio O. Introduction In this review I shall consider the perturbed N-body system, i.e., a system composed of N point bodies of masses ml, ... mN, described in cartesian co ordinates by the system of equations (0.1) where f) V'k,m == -GBPl--' m = 1, 2, 3.

Table of Contents

Classical Solutions for a Perturbed N-Body System.- Variational Setting for Newton's Equations.- The Kepler Problem Revisited.- The N-Body Problem.- Results form Critical Point Theory.- Classical Periodic Solutions for the Perturbed N-Body System.- Acknowledgments.- References.- Degree Theory: Old and New.- Degree Theory for Maps in the Sobolev Class H1(S2, S2).- Degree Theory for Maps in the Sobolev Class H1(S1, S1).- Degree Theory for Maps in VMO (Sn, Sn).- Further Properties of VMO Maps in Connection with Topology.- Degree Theory for VMO Maps on Domains.- References.- Global Structure for Nonlinear Operators in Differential and Integral Equations I. Folds.- Frechet Derivatives.- Fredholm Maps.- Local Structure of Folds.- Abstract Global Characterization of the Fold Map.- Ambrosetti-Prodi and Berger-Podolak - Church Fold Maps.- McKean-Scovel Fold Map.- Giannoni-Micheletti Fold Map.- Mandhyan Fold Map.- Oriented Global Fold Maps.- A Second Mandhyan Fold Map.- Jumping Singularities.- References.- Global Structure for Nonlinear Operators in Differential and Integral Equations II. Cusps.- Critical Values of Fredholm Maps.- Applications of Critical Values to Nonlinear Differential Equations.- Factorization of Differentiate Maps.- Local Structure of Cusps.- Some Local Cusp Results.- von Karman Equations.- Abstract Global Characterization of the Cusp Map.- Mandhyan Integral Operator Cusp Map.- Pseudo-Cusp.- Cafagna and Donati Theorems on Ordinary Differential Equations.- Micheletti Cusp-like Map.- Cafagna Dirichlet Example.- u3 Dirichlet Map - Initial Results.- u3 Dirichlet Map - The Singular Set and its Image.- u3 Dirichlet Map - The Global Result.- Ruf u3 Neumann Cusp Map.- Ruf's Higher Order Singularities.- Damon's Work in Differential Equations.- References.- Degree for Gradient Equivariant Maps and Equivariant Conley Index.- Basic Notions of Equivariant Topology.- Remarks and Examples.- An Analytic Definition of the Gradient Equivariant Degree.- Technicalities.- Equivariant Conley Index.- Box-like Index Pairs.- The torn Dieck Ring.- Bifurcation.- References.- Variations and Irregularities.- Summary.- Generalized Differential Operators.- Irregularities.- Mass, Length, Energy.- Homogeneous Dirichlet Spaces.- Fractals.- References.- Singularity Theory and Bifurcation Phenomena in Differential Equations.- The Normal Forms for f : ?n ? ?m.- The Malgrange Preparation Theorem.- Singularity Theory for Mappings Between Banach Spaces.- Applications to Elliptic Boundary Value Problems.- First Order Differential Equations.- Global Equivalence Theorems.- Problems with Additional Parameters: Unfoldings.- Bifurcation of Minimal Surfaces.- Singularities at Double Eigenvalues.- Multiplicity by combining Local and Global Information.- Some Numerical Results.- References.- Bifurcation from the Essential Spectrum.- General Setting.- Nonlinear Perturbation of a Self-Adjoint Operator.- Bifurcation from the Infimum of the Spectrum.- Bifurcation into Spectral Gaps.- Semilinear Elliptic Equations.- References.- Rotation of Vector Fields: Definition, Basic Properties, and Calculation.- The Brouwer-Hopf Theory of Continuous Vector Fields.- The Leray-Schauder Theory of Completely Continuous Vector Fields.- Vector Fields with Noncompact Operators.- Some Generalizations and Modifications.- References.
Volume

[1] : sz ISBN 9783764337421

Description

The main purpose of this text is to give a survey of some of the most significant achievements obtained by topological methods in nonlinear analysis during the last three decades. The book covers the three main branches of research in this field - topological degree, singularity theory and variational methods.

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