Hamilton-Jacobi Equations in Infinite Dimensions. Part 1. Uniqueness of Viscosity Solutions

Download or read book Hamilton-Jacobi Equations in Infinite Dimensions. Part 1. Uniqueness of Viscosity Solutions written by M. G. Crandall and published by . This book was released on 1984 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: The recent introduction of the theory of viscosity solutions of nonlinear first-order partial differential equations - which we will call Hamilton-Jacobi equations or HJE's here - has stimulated a very strong development of the existence and uniqueness theory of HJE's as well as a revitalization and perfection of the theory concerning the interaction between HJE's and the diverse areas in which they arise. The areas of application include the calculus of variations, control theory and differential games. This paper is the first of a series by the authors concerning the theoretical foundations of a corresponding program in infinite dimensional spaces. The basic question of what the appropriate notion of a viscosity solution should be in an infinite dimensional space is answered in spaces with the Radon-Nikodym property by observing that the finite dimensional characterization may be used essentially unchanged. Technical difficulties which arise in attempting to work with this definition because bounded continuous functions on balls in infinite dimensional spaces need not have maxima are dispatched with the aid of the variational principle which states that maxima do exist upon perturbation by an arbitrarily small linear functional.