Hamilton-Jacobi equations in infinite dimensions. I: Uniqueness of viscosity solutions. (English) Zbl 0627.49013

Crandall, Michael G.; Lions, Pierre-Louis

J. Funct. Anal. 62, 379-396 (1985).

In a series of previous papers the authors have introduced the concept of viscosity solutions for Hamilton-Jacobi equations in \({\mathbb{R}}^ n\) of the form \(H(x,u,Du)=0\) and proved existence and uniqueness for a large class of Hamiltonian functions H. This is a first work of a series of papers where the authors extend their beautiful theory to infinite dimensional Banach spaces with the Radon-Nikodym property. The main result of this paper is a uniqueness theorem of viscosity solutions under appropriate conditions on H which resemble those from the finite dimensional case. A key ingredient of the uniqueness in the infinite dimensional setting is Stegall’s variational principle.

Reviewer: V.Barbu

MSC:

49L99	Hamilton-Jacobi theories
35D05	Existence of generalized solutions of PDE (MSC2000)
35F20	Nonlinear first-order PDEs
46B99	Normed linear spaces and Banach spaces; Banach lattices

Keywords:

viscosity solutions; Hamilton-Jacobi equations; infinite dimensional Banach spaces with the Radon-Nikodym property

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