a b s t r a c tWe consider the following problem: given a bounded domain Ω ⊂ R n and a vector field ζ : Ω → R n , find a solution to −∆ ∞ u − ⟨Du, ζ ⟩ = 0 in Ω, u = f on ∂Ω, where ∆ ∞ is the 1-homogeneous infinity Laplace operator that is formally given by ∆ ∞ u = ⟨D 2 u Du |Du| , Du |Du| ⟩ 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-ofwar games we prove that this problem has a solution.
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