The infinity Laplacian with a transport term

We consider the following problem: given a bounded domain Ω ⊂ R n and a vector field ζ : Ω → R n , find a solution to − Δ ∞ u − 〈 D u , ζ 〉 = 0 in Ω , u = f on ∂ Ω , where Δ ∞ is the 1-homogeneous infinity Laplace operator that is formally given by Δ ∞ u = 〈 D 2 u D u | D u | , D u | D u | 〉 and f a Lipschitz boundary datum. If we assume that ζ is a continuous gradient vector field then we obtain the existence and uniqueness of a viscosity solution by an L p -approximation procedure. Also we prove the stability of the unique solution with respect to ζ . In addition when ζ is more regular (Lipschitz continuous) but not necessarily a gradient, using tug-of-war games we prove that this problem has a solution.