Hamilton-Jacobi Equations in Infinite Dimensions. Part 1. Uniqueness of Viscosity Solutions,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The recent introduction of the theory of viscosity solutions of nonlinear first-order partial differential equations - which we will call Hamilton-Jacobi equations or HJEs here - has stimulated a very strong development of the existence and uniqueness theory of HJEs as well as a revitalization and perfection of the theory concerning the interaction between HJEs 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.
- Numerical Mathematics
- Fluid Mechanics