Eikonal equations in metric spaces

Abstract

A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to infinite dimensional setting as well as topological networks, surfaces with singularities.

Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

“…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].…”

Eikonal equations on the Sierpinski gasket

“…In this section we show that, in analogy with the construction of the Laplacian on fractal sets, the eikonal equation on the Sierpinski gasket can be obtained as the limit of a sequence of discrete problems defined on the prefractal S n . We first recall some notations and definitions in [3]. Let (X , d) be a metric space.…”

“…The crucial point in [3] is a notion of metric derivative |ξ (t)| for a given path ξ = ξ(t) in X although in general ξ (t) may be not well defined. It follows that this elegant theory is confined to the case of the eikonal equation and difficult to extend to a more general class of Hamilton-Jacobi equation on S. Moreover the notion of viscosity supersolution in [3] is not local, even if consistent with the Euclidean viscosity solution. Instead we are able to extend the interior approximation approach to a more general class of Hamiltonians showing that the corresponding sequence of the solutions of the graph Hamilton-Jacobi equations converges uniformly to a function u on S. Lacking a definition of viscosity solution for general convex Hamiltonian on the Sierpinski gasket, our approach can be seen as a constructive way to define the solution to Hamilton-Jacobi equations on S. From this point of view the previous result for the eikonal equation can be also interpreted as a test that the construction gives the correct solution on the limit fractal.…”

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].…”