Viscosity Solutions of Hamilton-Jacobi Equations in Proper CAT (0) Spaces

In this article, we develop a novel notion of viscosity solutions for first order Hamilton-Jacobi equations in proper CAT (0) spaces. The notion of viscosity is defined by taking test functions that are locally Lipschitz and can be represented as a difference of two semiconvex functions. Under mild assumptions on the Hamiltonian, we recover the main features of viscosity theory for both the stationary and the time-dependent cases in this setting: the comparison principle and Perron’s method. Finally, we show that this notion of viscosity coincides with the classical one in R^N and we give several examples of Hamilton-Jacobi equations in more general CAT (0) spaces covered by this setting.