The infinity Laplacian with a transport term

Abstract

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.

Mass transport problems obtained as limits of p-Laplacian type problems with spatial dependence

“…Remark 2.3.Following the ideas in[7] it can be proved that a continuous weak solution to (1.1) is a viscosity solution to the same equation.Our next step is to prove that we can extract a sequence of solutions to (1.1), with → ∞, that converges uniformly as → ∞.Lemma 2.4.Let be a solution to (1.1), > . There exists a sequence → ∞ such that → ∞ uniformly in .…”

“…Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 7/12/15 2:24 PM When is of the form = − , > 0, for the study of the limit equation (in the viscosity sense) when → ∞ we refer to [7]. Here we focus our attention on the mass transport problem obtained in this limit procedure rather than in the equation that is veri ed by the limit.…”

An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term

“…The study of the singular inhomogeneous normalized infinity Laplace equation with a transport term (1) in this work was inspired by recent works, [23,34] and [2,3,4,10,20,24,38,39], on the normalized counterpart with a transport term by the game theory, and on the parabolic equations associated with infinity Laplacian by the partial differential equation theory.…”

“…In [23] the elliptic normalized infinity Laplacian with a transport term −∆ N ∞ u − ξ, Du = 0, in Ω, u = g, on ∂Ω (2) was first studied and the existence of viscosity solutions as the continuous value of the modified tug-of-war game is obtained by probabilistic approach when ξ is a Lipschitz vector field. While for a continuous gradient vector field ξ, the existence and uniqueness of viscosity solutions were obtained by the p−Laplace approximation.…”

“…While for a continuous gradient vector field ξ, the existence and uniqueness of viscosity solutions were obtained by the p−Laplace approximation. Let us briefly recall from [23] the two-player, random-turn, tug-of-war game with a transport term. Set G is the final payoff function defined in a narrow strip around the boundary ∂Ω.…”