An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞ PDF Download
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Author: Nikos Katzourakis
Publisher: Springer
ISBN: 3319128299
Category : Mathematics
Languages : en
Pages : 123
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Book Description
The purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.
Author: Nikos Katzourakis
Publisher: Springer
ISBN: 3319128299
Category : Mathematics
Languages : en
Pages : 123
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Book Description
The purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.
Author: N. V. Krylov
Publisher: American Mathematical Soc.
ISBN: 1470447401
Category : Differential equations, Parabolic
Languages : en
Pages : 441
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Book Description
This book concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. We look for solutions in Sobolev classes, local or global, or for viscosity solutions. Most of the auxiliary results, such as Aleksandrov's elliptic and parabolic estimates, the Krylov–Safonov and the Evans–Krylov theorems, are taken from old sources, and the main results were obtained in the last few years. Presentation of these results is based on a generalization of the Fefferman–Stein theorem, on Fang-Hua Lin's like estimates, and on the so-called “ersatz” existence theorems, saying that one can slightly modify “any” equation and get a “cut-off” equation that has solutions with bounded derivatives. These theorems allow us to prove the solvability in Sobolev classes for equations that are quite far from the ones which are convex or concave with respect to the Hessians of the unknown functions. In studying viscosity solutions, these theorems also allow us to deal with classical approximating solutions, thus avoiding sometimes heavy constructions from the usual theory of viscosity solutions.
Author: Nikolaos Katzourakis
Publisher: CRC Press
ISBN: 1351765329
Category : Mathematics
Languages : en
Pages : 378
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Book Description
Aimed primarily at undergraduate level university students, An Illustrative Introduction to Modern Analysis provides an accessible and lucid contemporary account of the fundamental principles of Mathematical Analysis. The themes treated include Metric Spaces, General Topology, Continuity, Completeness, Compactness, Measure Theory, Integration, Lebesgue Spaces, Hilbert Spaces, Banach Spaces, Linear Operators, Weak and Weak* Topologies. Suitable both for classroom use and independent reading, this book is ideal preparation for further study in research areas where a broad mathematical toolbox is required.
Author: Pablo Blanc
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110621797
Category : Mathematics
Languages : en
Pages : 234
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Book Description
Extending the well-known connection between classical linear potential theory and probability theory (through the interplay between harmonic functions and martingales) to the nonlinear case of tug-of-war games and their related partial differential equations, this unique book collects several results in this direction and puts them in an elementary perspective in a lucid and self-contained fashion.
Author: Martino Bardi
Publisher: Springer
ISBN: 9783662205976
Category : Mathematics
Languages : en
Pages : 266
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Book Description
The volume comprises five extended surveys on the recent theory of viscosity solutions of fully nonlinear partial differential equations, and some of its most relevant applications to optimal control theory for deterministic and stochastic systems, front propagation, geometric motions and mathematical finance. The volume forms a state-of-the-art reference on the subject of viscosity solutions, and the authors are among the most prominent specialists. Potential readers are researchers in nonlinear PDE's, systems theory, stochastic processes.
Author: Robert James Elliott
Publisher: Harlow, England : Longman Scientific & Technical
ISBN:
Category : Calculus of variations
Languages : en
Pages : 116
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Book Description
Author: Nikolaĭ Vladimirovich Krylov
Publisher:
ISBN: 9781470448530
Category : MATHEMATICS
Languages : en
Pages : 458
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Book Description
Author: Mikko Parviainen
Publisher: Springer Nature
ISBN: 9819978793
Category :
Languages : en
Pages : 83
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Book Description
Author: Wendell Helms Fleming
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 456
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Book Description
This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions.
Author: Yves Achdou
Publisher: Springer
ISBN: 3642364330
Category : Mathematics
Languages : en
Pages : 316
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Book Description
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).
