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.