Stability properties and large time behavior of viscosity solutions of Hamilton–Jacobi equations on metric spaces

Abstract

We investigate the asymptotic behavior of a solution of a Hamilton-Jacobi equation defined on a general metric space. Our results include general stability property and large time behavior of the solution. We argue from the viewpoint of viscosity solutions theory and adopt the notion of solution recently introduced by Gangbo and Święch to the Hamilton-Jacobi equation on a geodesic space.

Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

“…In conclusion, we see that u α converges to u locally uniformly on [0, ∞) × X. Since p α /α converges to p locally uniformly on R + , the stability property ( [30]) implies that u is a metric viscosity solution of (4.11).…”

“…Well-posedness of the Hamilton-Jacobi equation above in the framework of viscosity solutions has recently been established in a large class of metric spaces called geodesic spaces [3,11,14,12,29,30]; see also well-posedness results and applications on networks [34,1,18,35,16,17,7]. A metric space (X, d) is said to be geodesic if for any x, y ∈ X, there exists a geodesic γ t (t ∈ [0, 1]) in X joining x, y with a constant speed; in other words, we have γ 0 = x, γ 1 = y and d(γ s , γ t ) = |s − t|d(x, y) for any s, t ∈ [0, 1].…”

Hamilton-Jacobi in metric spaces with a homological term

“…Recently, the notion of viscosity solution of the Hamilton-Jacobi equation has been extended ( [1], [11], [12], [14] [21]) to a very general class of metric spaces. The definitions of [1], [11] and [14] are different; throughout the paper, we stick to the one of [11], which is used also in [21].…”

“…Recently, the notion of viscosity solution of the Hamilton-Jacobi equation has been extended ( [1], [11], [12], [14] [21]) to a very general class of metric spaces. The definitions of [1], [11] and [14] are different; throughout the paper, we stick to the one of [11], which is used also in [21]. It is also possible to extend the notion of solution to the viscous Hamilton-Jacobi equation, though in order to define the Laplacian we need the heavier structure of a metric measure space, i. e. a metric space (M, d) together with a Borel measure m. If the metric measure space (M, d, m) satisfies a very heavy hypothesis (which is called the RCD(K, ∞) condition) then the Laplacian and the heat flow on M are sufficiently well-behaved to replicate ( [6], [7]) the standard approach to the viscous Hamilton-Jacobi equation.…”

Equivalence of solutions of eikonal equation in metric spaces

“…This was extended to the class of potentially nonconvex Hamiltonians H by Gangbo and Święch [20,21], who proposed a generalized notion of viscosity solutions via appropriate test classes and proved uniqueness and existence of the solutions to more general Hamilton-Jacobi equations in length spaces. Stability and convexity of such solutions are studied respectively in [34] and in [30]. Since this definition of solutions is based on the local slope, we shall call them slope-based solutions (or s-solutions for short) below.…”