Antonio Ambrosetti - Böcker

This is an introduction to nonlinear functional analysis, in particular to those methods based on differential calculus in Banach spaces. It is in two parts; the first deals with the geometry of Banach spaces and includes a discussion of local and global inversion theorems for differentiable mappings. In the second part, the authors are more concerned with bifurcation theory, including the Hopf bifurcation. They include plenty of motivational and illustrative applications, which indeed provide much of the justification of nonlinear analysis. In particular, they discuss bifurcation problems arising from such areas as mechanics and fluid dynamics. The book is intended to accompany upper division courses for students of pure and applied mathematics and physics; exercises are consequently included.

Many problems in science and engineering are described by nonlinear differential equations, which can be notoriously difficult to solve. Through the interplay of topological and variational ideas, methods of nonlinear analysis are able to tackle such fundamental problems. This graduate text explains some of the key techniques in a way that will be appreciated by mathematicians, physicists and engineers. Starting from elementary tools of bifurcation theory and analysis, the authors cover a number of more modern topics from critical point theory to elliptic partial differential equations. A series of Appendices give convenient accounts of a variety of advanced topics that will introduce the reader to areas of current research. The book is amply illustrated and many chapters are rounded off with a set of exercises.

This self-contained textbook provides the basic, abstract tools used in nonlinear analysis. The text discusses key results such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, and more.

The first part of this book is mainly intended as a textbook for students at the Sophomore-Junior level, majoring in mathematics, engineering, or the sciences in general. The book includes the basic topics in Ordinary Differential Equations, normally taught at the undergraduate level, such as linear and nonlinear equations and systems, Bessel functions, Laplace transform, stability, etc. It is written with ample flexibility to make it appropriate either as a course stressing application, or a course stressing rigor and analytical thinking. It also offers sufficient material for a one-semester graduate course, covering topics such as phase plane analysis, oscillation, Sturm-Liouville equations, Euler-Lagrange equations in Calculus of Variations, first and second order linear PDE in 2D. There are substantial lists of exercises at the ends of the chapters. In this edition complete solutions to all even number problems are given in the back of the book.The 2nd edition also includes some new problems and examples. An effort has been made to make the material more suitable and self-contained for undergraduate students with minimal knowledge of Calculus. For example, a detailed review of matrices and determinants has been added to the chapter on systems of equations. The second edition also contains corrections of some misprints and errors in the first edition.

The second edition has been revised to correct minor errata, and features a number of carefully selected new exercises, together with more detailed explanations of some of the topics. A complete Solutions Manual, containing solutions to all the exercises published in the book, is available.

Contents: I. Ekeland: Some Variational Methods Arising from Mathematical Economics.- A. Mas-Colell: Four Lectures on the Differentiable Approach to General Equilibrium Theory.- J. Scheinkman: Dynamic General Equilibrium Models.- S. Zamir: Topics in Non Cooperative Game Theory.

This volume contains the notes of the lectures delivered at the CIME course GeometricAnalysis andPDEsduringtheweekofJune11-162007inCetraro (Cosenza). The school consisted in six courses held by M. Gursky (PDEs in Conformal Geometry), E. Lanconelli (Heat kernels in sub-Riemannian s- tings),A. Malchiodi(Concentration of solutions for some singularly perturbed Neumann problems), G. Tarantello (On some elliptic problems in the study of selfdual Chern-Simons vortices), X. J. Wang (Thek-Hessian Equation)and P. Yang (Minimal Surfaces in CR Geometry). Geometric PDEs are a ?eld of research which is currently very active, as it makes it possible to treat classical problems in geometry and has had a dramatic impact on the comprehension of three- and four-dimensional ma- folds in the last several years. On one hand the geometric structure of these PDEs might cause general di?culties due to the presence of some invariance (translations, dilations, choice of gauge, etc. ), which results in a lack of c- pactness of the functional embeddings for the spaces of functions associated with the problems.On the other hand, a geometric intuition or result might contribute enormously to the search for natural quantities to keep track of, andtoproveregularityoraprioriestimatesonsolutions. Thistwo-foldaspect of the study makes it both challenging and complex, and requires the use of severalre?nedtechniquestoovercomethemajordi?cultiesencountered.

Several important problems arising in Physics, Di?erential Geometry and other n topics lead to consider semilinear variational elliptic equations on R and a great deal of work has been devoted to their study. From the mathematical point of view, the main interest relies on the fact that the tools of Nonlinear Functional Analysis, based on compactness arguments, in general cannot be used, at least in a straightforward way, and some new techniques have to be developed. n On the other hand, there are several elliptic problems on R which are p- turbative in nature. In some cases there is a natural perturbation parameter, like inthe bifurcationfromthe essentialspectrum orinsingularlyperturbed equations or in the study of semiclassical standing waves for NLS. In some other circ- stances, one studies perturbations either because this is the ?rst step to obtain global results or else because it often provides a correct perspective for further global studies. For these perturbation problems a speci?c approach,that takes advantage of such a perturbative setting, seems the most appropriate. These abstract tools are provided by perturbation methods in critical point theory. Actually, it turns out that such a framework can be used to handle a large variety of equations, usually considered di?erent in nature. Theaimofthismonographistodiscusstheseabstractmethodstogetherwith their applications to several perturbation problems, whose common feature is to n involve semilinear Elliptic Partial Di?erential Equations on R with a variational structure.

Le equazioni differenziali sono un argomento fondamentale non solo della matematica, ma anche della fisica, dell’ingegneria e, in generale, di tutte le scienze. Questo volume intende fornire allo studente una panoramica di alcune tra le più interessanti e suggestive questioni relative alle equazioni differenziali ordinarie trattate da un punto di vista geometrico, aprendo uno sguardo verso l’analisi funzionale. Oltre ai risultati classici sulle equazioni lineari, molto spazio è dato ai problemi nonlineari che spesso non sono oggetto dei corsi istituzionali.L'esposizione è tenuta a un livello semplice in modo che il libro possa essere accessibile a studenti dell’ultimo anno della laurea triennale e della laurea magistrale, offrendo anche spunti per ulteriori approfondimenti.

In this volume, various ideas about Hamiltonian dynamics were discussed. Particular emphasis was placed on mechanical systems with singular potentials (such as the N-Body Newtonian problem) and on their special features, although important aspects of smooth dynamics were also discussed, from both the local point of view and the point of view of global analysis.