We propose a novel notion of viscosity solutions to study first order Hamilton-Jacobi equa- tions in a certain class of metric spaces called geodesic spaces of one curvature bound in the sense of Alexandrov.
A metric space (X,d) is said to be a geodesic space of curvature bounded from above (re- spectively, from below) in the sense of Alexandrov if, roughly speaking, it is a geodesic space and of curvature bounded from below (respectively bounded from above) in the sense of the trian- gle comparison theorem. They can be seen as a generalization of Riemannian manifolds with sectional curvature bounded from below (respectively bounded from above). Typical examples include Hilbert spaces, connected Riemannian manifolds that have everywhere sectional curva- ture bounded from above or from below, metric trees, compact subsets of positive reach in the Euclidean space and the quadratic Wasserstein space over the Euclidean space. In the literature, geodesic spaces of curvature bounded from below in the sense of Alexandrov are called Alexan- drov spaces. On the other hand, geodesic spaces of curvature bounded from above in the sense of Alexandrov are called CAT(κ) spaces, where κ ∈ R refers to the curvature bound, and the acronym CAT refers to the names of the three mathematicians É. Cartan, A.D. Alexandrov and V.A. Topono- gov who were the first to describe curvature using inequalities involving the distance d of a metric space.
Although these types of metric 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 a differential is well defined for any real-valued function u : X → R that is locally Lipschitz and can be represented as a difference of two semiconvex functions (Lipschitz and DC functions in short). We propose to exploit all this structure that these 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 will be mainly interested in CAT(0) spaces and Wasserstein spaces over a com- pact Riemannian manifold. 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 = R^N .