The purpose of this work is twofold. First we study the solutions of a Hamilton-Jacobi equation of the form $u_t(t,x)+\mathcal{ H}(t,x,\nabla_H u(t,x))=0$, where $\nabla_H u$ represents the horizontal gradient of a function $u$ defined on the Heisenberg group ${ I\!\!H}$. Motivated by the recent paper by Liu, Manfredi and Zhou (\cite{ LiMaZh2016}), we prove a Lipschitz continuity preserving property for $u$ with respect to the Kor\'anyi homogeneous distances $d_G$ in ${ I\!\!H}$. Secondly, we are keenly interested in introducing the game theory in ${ I\!\!H}$, taking into account its Sub-Riemannian structure: inspired by ideas in the paper of Evans and Souganidis (see \cite{ EvSo1984}), in the paper of and Balogh, Calogero and Pini \cite{ BaCaPi2014}, we prove $d_G$-Lipschitz regularity results for the lower and the upper value functions of a zero game with horizontal curves as its trajectories, and we study the Hamilton-Jacobi-Isaacs equations associated to such zero game. As a consequence, we also provide a representation of the viscosity solution of the initial Hamilton-Jacobi equation.
Categories: math.AP