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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. Key features of this book: * Self-contained in-depth treatment of sign-changing critical point theory * Further explorations in minimax and Morse theories * Topics devoted to linking and nodal solutions, the sign-changing saddle point theory, the generalized Brezis–Nirenberg critical point theorem, the parameter dependence of sign-changing critical points * Applications of sign-changing critical point theory studied within the classical symmetric mountain pass theorem *Applies sign-changing concepts to Schrödinger equations and boundary value problems This book is intended for advanced graduate students and researchers involved in sign-changing critical point theory, PDEs, global analysis, and nonlinear functional analysis. Also by the author: (with Martin Schechter) Critical Point Theory and Its Applications, ©2006, Springer, ISBN: 978-0-387-32965-9.

Mathematics. --- Approximations and Expansions. --- Topology. --- Functional Analysis. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Global Analysis and Analysis on Manifolds. --- Functional analysis. --- Global analysis. --- Differential equations, partial. --- Mathematical optimization. --- Mathématiques --- Analyse fonctionnelle --- Optimisation mathématique --- Topologie --- Critical point theory (Mathematical analysis). --- Mathematical analysis. --- Critical point theory (Mathematical analysis) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Global analysis (Mathematics) --- Analysis, Global (Mathematics) --- Approximation theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Partial differential equations. --- Calculus of variations. --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Calculus of variations --- Math --- Science --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Functional calculus --- Functional equations --- Integral equations --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Isoperimetrical problems --- Variations, Calculus of --- Geometry, Differential --- Topology

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This book gives for the first time a self-contained and unified approach to holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel on manifolds by using the heat kernel, and presents also various applications. The main analytic tool is the analytic localization technique in local index theory developed by Bismut-Lebeau. The book includes the most recent results in the field and therefore opens perspectives on several active areas of research in complex, Kähler and symplectic geometry. A large number of applications are included, e.g., an analytic proof of the Kodaira embedding theorem, a solution of the Grauert-Riemenschneider and Shiffman conjectures, a compactification of complete Kähler manifolds of pinched negative curvature, the Berezin-Toeplitz quantization, weak Lefschetz theorems, and the asymptotics of the Ray-Singer analytic torsion.

Bergman kernel functions. --- Holomorphic functions. --- Morse theory. --- Symplectic manifolds. --- Variational inequalities (Mathematics) --- Inequalities, Variational (Mathematics) --- Calculus of variations --- Differential inequalities --- Functions, Holomorphic --- Functions of several complex variables --- Kernel functions, Bergman --- Holomorphic mappings --- Kernel functions --- Manifolds, Symplectic --- Geometry, Differential --- Manifolds (Mathematics) --- Critical point theory (Mathematical analysis) --- Morse, théorie de --- Inégalités variationnelles --- Fonctions holomorphes --- Variétés symplectiques --- EPUB-LIV-FT SPRINGER-B LIVMATHE --- Global differential geometry. --- Differential equations, partial. --- Global analysis. --- Differential Geometry. --- Several Complex Variables and Analytic Spaces. --- Global Analysis and Analysis on Manifolds. --- Global analysis (Mathematics) --- Partial differential equations --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential geometry. --- Functions of complex variables. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Complex variables --- Elliptic functions --- Functions of real variables --- Topology --- Differential geometry