Read more
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.
List of contents
Preface.- Linking Systems.- Sandwich Systems.- Linking Sandwich Sets.- The Monotonicity Trick.- Infinite Dimensional Linking.- Differential Equations.- Schrödinger Equations.- Zero in the Spectrum.- Global Solutions.- Second Order Hamiltonian Systems.- Core Functions.- Custom Monotonicity Methods.- Elliptic Systems.- Flows and Critical Points.- The Semilinear Wave Equation.- Nonlinear Optics.- Radially Symmetric Wave Equations.- Multiple Solutions.
Report
"Many problems arizing in science and engineering call for the solving of nonlinear ordinary differential equations or partial differential equations. These equations are difficult to solve, and there are few general techniques ... to solve them. ... This has motivated researchers to study critical points of functionals in order to solve the corresponding Euler equations. It has led to the development of several techniques to find critical points. This book is dedicated to the latest developments and applications of these techniques." (Mohsen Timoumi,zbMATH 1462.35008, 2021)
