The infinity Laplacian with a transport term

We consider the following problem: given a bounded domain Ω⊂RnΩ⊂Rn and a vector field ζ:Ω→Rnζ:Ω→Rn, find a solution to −Δ∞u−〈Du,ζ〉=0−Δ∞u−〈Du,ζ〉=0 in ΩΩ, u=fu=f on ∂Ω∂Ω, where Δ∞Δ∞ is the 1-homogeneous infinity Laplace operator that is formally given by Δ∞u=〈D2uDu|Du|,Du|Du|〉 and ff a Lipschitz boundary datum. If we assume that ζζ is a continuous gradient vector field then we obtain the existence and uniqueness of a viscosity solution by an LpLp-approximation procedure. Also we prove the stability of the unique solution with respect to ζζ. In addition when ζζ is more regular (Lipschitz continuous) but not necessarily a gradient, using tug-of-war games we prove that this problem has a solution.