Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais-Smale condition. The existence of the bifurcation branch and the asymptotic nature of the bifurcation is verified by using the compactness in the Palais Smale condition and the order of the nonlinearity in the operator. The main result is applied to estimate the asyptotic behaviour of solutions to a class of semilinear elliptic equations with a critical Sobolev exponent

Variational Methods for NLEV Approximation Near a Bifurcation Point

We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains ofℝN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included

Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations

Coupled nonlinear Schrodinger equations (CNLS) govern many physical phenomena, such as nonlinear optics and Bose-Einstein condensates. For their wide applications, many studies have been carried out by physicists, mathematicians and engineers from different respects. In this dissertation, we focused on standing wave solutions, which are of particular interests for their relatively simple form and the important roles they play in studying other wave solutions. We studied the multiplicity of this type of solutions of CNLS via variational methods and bifurcation methods. Variational methods are useful tools for studying differential equations and systems of differential equations that possess the so-called variational structure. For such an equation or system, a weak solution can be found through finding the critical point of a corresponding energy functional. If this equation or system is also invariant under a certain symmetric group, multiple solutions are often expected. In this work, an integer-valued function that measures symmetries of CNLS was used to determine critical values. Besides variational methods, bifurcation methods may also be used to find solutions of a differential equation or system, if some trivial solution branch exists and the system is degenerate somewhere on this branch. If local bifurcations exist, then new solutions can be found in a neighborhood of each bifurcation point. If global bifurcation branches exist, then there is a continuous solution branch emanating from each bifurcation point. We consider two types of CNLS. First, for a fully symmetric system, we introduce a new index and use it to construct a sequence of critical energy levels. Using variational methods and the symmetric structure, we prove that there is at least one solution on each one of these critical energy levels. Second, we study the bifurcation phenomena of a two-equation asymmetric system. All these bifurcations take place with respect to a positive solution branch that is already known. The locations of the bifurcation points are determined through an equation of a coupling parameter. A few nonexistence results of positive solutions are also give

Foundations of Mechanics, Second Edition

Preface to the Second Edition. Since the first edition of this book appeared in 1967, there has been a great deal of activity in the field of symplectic geometry and Hamiltonian systems. In addition to the recent textbooks of Arnold, Arnold-Avez, Godbillon, Guillemin-Sternberg, Siegel-Moser, and Souriau, there have been many research articles published. Two good collections are "Symposia Mathematica," vol. XIV, and "Géométrie Symplectique el Physique Mathématique," CNRS, Colloque Internationaux, no. 237. There are also important survey articles, such as Weinstein [1977b]. The text and bibliography contain many of the important new references we are aware of. We have continued to find the classic works, especially Whittaker [1959], invaluable. The basic audience for the book remains the same: mathematicians, physicists, and engineers interested in geometrical methods in mechanics, assuming a background in calculus, linear algebra, some classical analysis, and point set topology. We include most of the basic results in manifold theory, as well as some key facts from point set topology and Lie group theory. Other things used without proof are clearly noted. We have updated the material on symmetry groups and qualitative theory, added new sections on the rigid body, topology and mechanics, and quantization, and other topics, and have made numerous corrections and additions. In fact, some of the results in this edition are new. We have made two major changes in notation: we now use f^* for pull-back (the first edition used f[sub]*), in accordance with standard usage, and have adopted the "Bourbaki" convention for wedge product. The latter eliminates many annoying factors of 2. A. N. Kolmogorov's address at the 1954 International Congress of Mathematicians marked an important historical point in the development of the theory, and is reproduced as an appendix. The work of Kolmogorov, Arnold, and Moser and its application to Laplace's question of stability of the solar system remains one of the goals of the exposition. For complete details of all tbe theorems needed in this direction, outside references will have to be consulted, such as Siegel-Moser [1971] and Moser [1973a]. We are pleased to acknowledge valuable assistance from Paul Chernoff, Wlodek Tulczyjew, Morris Hirsh, Alan Weinstein, and our invaluable assistant authors, Richard Cushman and Tudor Ratiu, who all contributed some of their original material for incorporation into the text. Also, we are grateful to Ethan Akin, Kentaro Mikami, Judy Arms, Harold Naparst, Michael Buchner, Ed Nelson, Robert Cahn, Sheldon Newhouse, Emil Chorosoff, George Oster, André Deprit, Jean-Paul Penot, Bob Devaney, Joel Robbin, Hans Duistermaat, Clark Robinson, John Guckenheimer, David Rod, Martin Gutzwiller, William Satzer, Richard Hansen, Dieter Schmidt, Morris Kirsch, Mike Shub, Michael Hoffman, Steve Smale, Andrei Iacob, Rich Spencer, Robert Jantzen, Mike Spivak, Therese Langer, Dan Sunday, Ken Meyer, Floris Takens, [and] Randy Wohl for contributions, remarks, and corrections which we have included in this edition. Further, we express our gratitude to Chris Shaw, who made exceptional efforts to transfom our sketches into the graphics which illustrate the text, to Peter Coha for his assistance in organizing the Museum and Bibliography, and to Ruthie Cephas, Jody Hilbun, Marnie McElhiney, Ruth (Bionic Fingers) Suzuki, and Ikuko Workman for their superb typing job. Theoretical mechanics is an ever-expanding subject. We will appreciate comments from readers regarding new results and shortcomings in this edition. RALPH ABRAHAM, JERROLD E. MARSDEN</p

Causal spaces and the application of critical point theory to general relativity

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Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute

Computer-assisted Multiplicity Proofs for Emden\u27s Equation on Domains with Hole

In this thesis we consider Emden\u27s equation on a parameter dependent domain with hole and describe a computer-assisted method for proving existence and multiplicity of solutions to that problem. We obtain results for discrete parameter values as well as for some parameter intervals, in which case also the existence of smooth solution branches is proved. Moreover we prove the existence of a one-bump solution to Emden\u27s equation on an unbounded L-shaped domain