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This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L -Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L -Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the “metric” theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to Non-Smooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability.

Measure theory. --- Metric spaces. --- Differential equations, Parabolic. --- Monotone operators. --- Evolution equations, Nonlinear. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Nonlinear equations of evolution --- Nonlinear evolution equations --- Differential equations, Nonlinear --- Operator theory --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Spaces, Metric --- Generalized spaces --- Set theory --- Topology --- Mathematics. --- Global differential geometry. --- Distribution (Probability theory. --- Measure and Integration. --- Differential Geometry. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Geometry, Differential --- Math --- Science --- Differential geometry. --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Differential geometry

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Measure theory. --- Metric spaces. --- Differential equations, Parabolic. --- Evolution equations, Nonlinear.

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Differential geometry. Global analysis --- Operational research. Game theory --- Mathematical physics --- differentiaalvergelijkingen --- differentiaal geometrie --- stochastische analyse --- kansrekening

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This volume collects the notes of the CIME course "Nonlinear PDE’s and applications" held in Cetraro (Italy) on June 23–28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of far-reaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, Hamilton-Jacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rate-independent evolution problems. The book explores the fundamental connections between all of these topics and points to new research directions in contributions by leading experts in these fields.

Differential equations, Partial --- Differential equations, Nonlinear --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Calculus --- Applied Mathematics --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Partial differential equations. --- Differential geometry. --- Calculus of variations. --- Analysis. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Functional Analysis. --- Differential Geometry. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Differential geometry --- Partial differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Global analysis (Mathematics). --- Differential equations, partial. --- Mathematical optimization. --- Global differential geometry. --- Geometry, Differential --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Operations research --- Simulation methods --- System analysis --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Nonlinear PDEs --- CIME

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Teoria de la mesura geomètrica --- Càlcul de variacions --- Equacions en derivades parcials --- Geometria diferencial --- Geometria --- Càlcul de tensors --- Connexions (Matemàtica) --- Coordenades --- Corbes --- Cossos convexos --- Dominis convexos --- Espais de curvatura constant --- Espais simètrics --- Estructures hermitianes --- Formes diferencials --- G-estructures --- Geodèsiques (Matemàtica) --- Geometria de Riemann --- Geometria diferencial global --- Geometria integral --- Geometria simplèctica --- Hiperespai --- Subvarietats (Matemàtica) --- Topologia diferencial --- Varietats (Matemàtica) --- Varietats de Kähler --- EDPs --- Equació diferencial en derivades parcials --- Equacions diferencials en derivades parcials --- Equacions diferencials parcials --- Equacions diferencials --- Dispersió (Matemàtica) --- Equació d'ona --- Equació de Dirac --- Equació de Fokker-Planck --- Equació de Schrödinger --- Equacions de Navier-Stokes --- Equacions de Hamilton-Jacobi --- Equacions de Maxwell --- Equacions de Monge-Ampère --- Equacions de Von Kármán --- Equacions diferencials el·líptiques --- Equacions diferencials hiperbòliques --- Equacions diferencials parabòliques --- Equacions diferencials parcials estocàstiques --- Funcions harmòniques --- Laplacià --- Problema de Cauchy --- Problema de Neumann --- Teoria espectral (Matemàtica) --- Càlcul variacional --- Problemes isoperimètrics --- Màxims i mínims --- Anàlisi funcional --- Desigualtats variacionals (Matemàtica) --- Funcions de Lagrange --- Principis variacionals --- Teoria de Morse --- Teoria del punt crític (Anàlisi matemàtica) --- Teoria de la mesura --- Calculus of variations. --- Geometry, Differential. --- Differential equations, Partial. --- Partial differential equations --- Differential geometry --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima