Duality and Perturbation Methods in Critical Point Theory PDF Download
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Author: Nassif Ghoussoub
Publisher: Cambridge University Press
ISBN: 9780521440257
Category : Mathematics
Languages : en
Pages : 358
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Book Description
The calculus of variations has been an active area of mathematics for over 300 years. Its main use is to find stable critical points of functions for the solution of problems. To find unstable values, new approaches (Morse theory and min-max methods) were developed, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a point of view that may help when dealing with such problems. Building upon min-max methods, he systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book reasonably self-contained. Consequently, it should be accessible to all mathematicians, pure or applied, economists and engineers working in nonlinear analysis or optimization.
Author: Nassif Ghoussoub
Publisher: Cambridge University Press
ISBN: 9780521440257
Category : Mathematics
Languages : en
Pages : 358
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Book Description
The calculus of variations has been an active area of mathematics for over 300 years. Its main use is to find stable critical points of functions for the solution of problems. To find unstable values, new approaches (Morse theory and min-max methods) were developed, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a point of view that may help when dealing with such problems. Building upon min-max methods, he systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book reasonably self-contained. Consequently, it should be accessible to all mathematicians, pure or applied, economists and engineers working in nonlinear analysis or optimization.
Author: Martin Schechter
Publisher: Springer Science & Business Media
ISBN: 146121596X
Category : Mathematics
Languages : en
Pages : 305
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Book Description
As is well known, The Great Divide (a.k.a. The Continental Divide) is formed by the Rocky Mountains stretching from north to south across North America. It creates a virtual "stone wall" so high that wind, rain, snow, etc. cannot cross it. This keeps the weather distinct on both sides. Since railroad trains cannot climb steep grades and tunnels through these mountains are almost formidable, the Canadian Pacific Railroad searched for a mountain pass providing the lowest grade for its tracks. Employees discovered a suitable mountain pass, called the Kicking Horse Pass, el. 5404 ft., near Banff, Alberta. (One can speculate as to the reason for the name.) This pass is also used by the Trans-Canada Highway. At the highest point of the pass the railroad tracks are horizontal with mountains rising on both sides. A mountain stream divides into two branches, one flowing into the Atlantic Ocean and the other into the Pacific. One can literally stand (as the author did) with one foot in the Atlantic Ocean and the other in the Pacific. The author has observed many mountain passes in the Rocky Mountains and Alps. What connections do mountain passes have with nonlinear partial dif ferential equations? To find out, read on ...
Author: Martin Schechter
Publisher: Springer Nature
ISBN: 303045603X
Category : Mathematics
Languages : en
Pages : 347
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Book Description
This monograph collects cutting-edge results and techniques for solving nonlinear partial differential equations using critical points. Including many of the author’s own contributions, a range of proofs are conveniently collected here, Because the material is approached with rigor, this book will serve as an invaluable resource for exploring recent developments in this active area of research, as well as the numerous ways in which critical point theory can be applied. Different methods for finding critical points are presented in the first six chapters. The specific situations in which these methods are applicable is explained in detail. Focus then shifts toward the book’s main subject: applications to problems in mathematics and physics. These include topics such as Schrödinger equations, Hamiltonian systems, elliptic systems, nonlinear wave equations, nonlinear optics, semilinear PDEs, boundary value problems, and equations with multiple solutions. Readers will find this collection of applications convenient and thorough, with detailed proofs appearing throughout. Critical Point Theory will be ideal for graduate students and researchers interested in solving differential equations, and for those studying variational methods. An understanding of fundamental mathematical analysis is assumed. In particular, the basic properties of Hilbert and Banach spaces are used.
Author: Kanishka Perera
Publisher: Cambridge University Press
ISBN: 110702966X
Category : Mathematics
Languages : en
Pages : 171
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Book Description
Provides an introduction to critical point theory and shows how it solves many difficult problems.
Author: Wenming Zou
Publisher: Springer Science & Business Media
ISBN: 0387766588
Category : Mathematics
Languages : en
Pages : 288
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Book Description
Many nonlinear problems in physics, engineering, biology and social sciences can be reduced to finding critical points of functionals. While minimax and Morse theories provide answers to many situations and problems on the existence of multiple critical points of a functional, they often cannot provide much-needed additional properties of these critical points. Sign-changing critical point theory has emerged as a new area of rich research on critical points of a differentiable functional with important applications to nonlinear elliptic PDEs. This book is intended for advanced graduate students and researchers involved in sign-changing critical point theory, PDEs, global analysis, and nonlinear functional analysis.
Author: Martin Schechter
Publisher: Springer Science & Business Media
ISBN: 0817649026
Category : Mathematics
Languages : en
Pages : 239
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Book Description
This text starts at the foundations of the field, and is accessible with some background in functional analysis. As such, the book is ideal for classroom of self study. The new material covered also makes this book a must read for researchers in the theory of critical points.
Author: Wenming Zou
Publisher: Springer Science & Business Media
ISBN: 0387329684
Category : Mathematics
Languages : en
Pages : 323
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Book Description
This book presents some of the latest research in critical point theory, describing methods and presenting the newest applications. Coverage includes extrema, even valued functionals, weak and double linking, sign changing solutions, Morse inequalities, and cohomology groups. Applications described include Hamiltonian systems, Schrödinger equations and systems, jumping nonlinearities, elliptic equations and systems, superlinear problems and beam equations.
Author: Nassif Ghoussoub
Publisher: Springer Science & Business Media
ISBN: 0387848975
Category : Mathematics
Languages : en
Pages : 352
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Book Description
This text is intended for a beginning graduate course on convexity methods for PDEs. The generality chosen by the author puts this under the classification of "functional analysis". The book contains new results and plenty of examples and exercises.
Author: Nikolaos S. Papageorgiou
Publisher: Springer
ISBN: 3030034305
Category : Mathematics
Languages : en
Pages : 586
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Book Description
This book emphasizes those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The content is developed over six chapters, providing a thorough introduction to the techniques used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations as well as their applications to various processes arising in the applied sciences. They show how these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book a nice balance is maintained between rigorous mathematics and physical applications. The primary readership includes graduate students and researchers in pure and applied nonlinear analysis.
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Languages : en
Pages : 224
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Book Description
