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These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).

Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Calculus --- Operations Research --- Mathematics. --- Difference equations. --- Functional equations. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Game theory. --- Computer mathematics. --- Calculus of variations. --- Calculus of Variations and Optimal Control; Optimization. --- Partial Differential Equations. --- Computational Mathematics and Numerical Analysis. --- Game Theory, Economics, Social and Behav. Sciences. --- Dynamical Systems and Ergodic Theory. --- Difference and Functional Equations. --- Viscosity solutions --- Approximation theory --- Idempotents --- Hamilton-Jacobi equations --- Numerical analysis --- Equations, Hamilton-Jacobi --- Equations, Jacobi-Hamilton --- Jacobi-Hamilton equations --- Calculus of variations --- Differential equations, Partial --- Hamiltonian systems --- Mechanics --- Idempotent elements --- Algebras, Linear --- Mathematical physics --- Mathematical optimization. --- Differential equations, partial. --- Computer science --- Differentiable dynamical systems. --- Equations, Functional --- Functional analysis --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Math --- Science --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Ergodic transformations --- Continuous groups --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Physics --- Statics --- Games, Theory of --- Theory of games --- Mathematical models --- Isoperimetrical problems --- Variations, Calculus of --- Differential equations. --- Mathematics—Data processing. --- Dynamical systems. --- Calculus of Variations and Optimization. --- Differential Equations. --- Game Theory. --- Dynamical Systems. --- 517.91 Differential equations

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Mathematical analysis --- Numerical methods of optimisation --- Operational research. Game theory --- Probability theory --- Mathematics --- Computer. Automation --- Recreation. Games. Sports. Corp. expression --- analyse (wiskunde) --- waarschijnlijkheidstheorie --- computers --- spellen --- wiskunde

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This volume provides an introduction to the theory of Mean Field Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a mean-field model for Nash equilibria in the strategic interaction of a large number of agents. Besides giving an accessible presentation of the main features of mean-field game theory, the volume offers an overview of recent developments which explore several important directions: from partial differential equations to stochastic analysis, from the calculus of variations to modeling and aspects related to numerical methods. Arising from the CIME Summer School "Mean Field Games" held in Cetraro in 2019, this book collects together lecture notes prepared by Y. Achdou (with M. Laurière), P. Cardaliaguet, F. Delarue, A. Porretta and F. Santambrogio. These notes will be valuable for researchers and advanced graduate students who wish to approach this theory and explore its connections with several different fields in mathematics.

Mathematical analysis --- Numerical methods of optimisation --- Operational research. Game theory --- Probability theory --- Mathematics --- Computer. Automation --- Recreation. Games. Sports. Corp. expression --- analyse (wiskunde) --- waarschijnlijkheidstheorie --- computers --- spellen --- wiskunde