A maximum principle for evolution Hamilton–Jacobi equations on Riemannian manifolds

Abstract

We establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form u t + F (t, d x u) = 0, u(0, x) = u 0 (x), where u 0 : M → R is a bounded uniformly continuous function, M is a Riemannian manifold, and F : [0, ∞) × T * M → R. This yields uniqueness of the viscosity solutions of such Hamilton-Jacobi equations.First order Hamilton-Jacobi equations are partial differential equations of the form F x, u(x), du(x) = 0 in the stationary case, and of the form, F t, x, u(x, t), du(t, x) = 0 in the evolution case. Such equations arise, for instance, in optimal control theory and differential games. It is very well known that, even in the simplest case where u : R → R, there are examples

A class of Hamilton–Jacobi equations on Banach–Finsler manifolds

“…Definition 5.2. The Hamiltonian H of (E3) satisfies condition (A) whenever there are a constant C ≥ 0 and a continuous function ω : [0, ∞) × R × R → R with ω(0, 0, 0) = 0 such that for any t 1 , t 2 ∈ [0, ∞), x 1 , x 2 ∈ M and r 1 , r 2 ∈ R, In the next result we follow the ideas of [5], [13,Theorem 6.2], [3], [12], [16] and [19] to obtain a generalization for Finsler manifolds.…”

“…In the next result we follow the ideas of [5], [13, Theorem 6.2], [3], [12], [16] and [19] to obtain a generalization for Finsler manifolds.…”

“…The study of the above mentioned concepts in (finite and infinite dimensional) Riemannian manifolds was introduced by D. Azagra, J. Ferrera and F. López-Mesas in [2,3]. Let us also mention the related work of Y.S.…”

“…This paper is a continuation of [20], where basic properties and a smooth variational principle were studied in the context of Banach-Finsler manifolds. In particular, we apply some of the results obtained in [20], as well as some techniques studied in the cases of Hamilton-Jacobi equations on R n , on Banach spaces and on Riemannian manifolds [18,19,14,10,13,2,3], in order to obtain our results about existence and uniqueness of viscosity solutions of a class of Hamilton-Jacobi equations on Banach-Finsler manifolds.…”