The infinity Laplacian with a transport term

We consider the following problem: given a bounded domain Ω⊂Rn and a vector field ζ:Ω→Rn, find a solution to −Δ∞u−〈Du,ζ〉=0 in Ω, u=f on ∂Ω, where Δ∞ is the 1-homogeneous infinity Laplace operator that is formally given by Δ∞u=〈D2uDu|Du|,Du|Du|〉 and f a Lipschitz boundary datum. If we assume that ζ i...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2013-02, Vol.398 (2), p.752-765
Main Authors: López-Soriano, Rafael, Navarro-Climent, José C., Rossi, Julio D.
Format: Article
Language:English
Subjects:
Gradient terms
Infinity Laplacian
Tug of war games
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Summary:We consider the following problem: given a bounded domain Ω⊂Rn and a vector field ζ:Ω→Rn, find a solution to −Δ∞u−〈Du,ζ〉=0 in Ω, u=f on ∂Ω, where Δ∞ is the 1-homogeneous infinity Laplace operator that is formally given by Δ∞u=〈D2uDu|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 Lp-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.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2012.09.030