We propose a novel viscosity notion to study first order Hamilton-Jacobi equations in a
certain class of metric spaces called proper CAT(0) spaces.
A metric space (X, d) is said to be a CAT(0) space if, roughly speaking, it is a geodesic
space and of non-positive curvature in the sense of triangle comparison theorem. They can be
seen as a generalization of Hilbert spaces or Hadamard manifolds. Typical examples of CAT(0)
spaces include Hilbert spaces, metric trees and networks obtained by gluing a finite number of
half-spaces along their common boundary.
Although CAT(0) spaces are not manifolds in general, they carry a solid first order differential
calculus resembling that of a Hilbert space. For example, a notion of tangent cone is well
defined at each point of X. The tangent cone is the metric counterpart of the tangent space in
Riemannian geometry or the Bouligand tangent cone in convex analysis. Furthermore, a notion
of differential is well defined for any real-valued function u : X → R that is Lipschitz and can be
represented as a difference of two semiconvex functions (Lipschitz and DC functions in short).
We propose to exploit all this additional structure that CAT(0) spaces enjoy to study stationary
and time dependent first order Hamilton-Jacobi equation in them. In particular, we want to
recover the main features of viscosity theory: the comparison principle and Perron’s method.
In this talk, we give the main hypotheses we require for the Hamiltonian in this setting.
Furthermore, we define the notion of viscosity solutions, using test functions that are Lipschitz
and DC. Moreover, we show that we obtain the comparison principle using the variable doubling
technique. Finally, we derive existence of the solution from the comparison principle using
Perron’s method in a similar manner as in the classical case of X = RN .
In the end, we will briefly explain how this novel notion of viscosity could be adopted in other
classes of geodesic spaces such as Wasserstein spaces over the Euclidean space or over compact
Riemannian manifolds, which can be regarded as geodesics space with non-negative curvature
in the sense of the triangle comparison theorem.
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