Uniqueness of ∞-Harmonic Functions and the Eikonal Equation

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A mixed problem for the infinity Laplacian via Tug-of-War games (2007, background): “When ΓD ≡ ∂Ω, it is known that problem (1) has a unique viscosity solution, (as proved in [12]; see also [5], [8], and in a more general framework, [19]).”
A finite difference approach to the infinity Laplace equation and tug-of-war games (2009, methodology): “There exist solutions u, ū ∈ C(Ω̄) ∩ C0,1(Ω) of (2.22) such that whenever w ∈ USC(Ω̄) (w ∈ LSC(Ω̄)) is a subsolution (supersolution) of the equation −∆∞w = f and w ≤ g (w ≥ g) on ∂Ω, we have w ≤ ū (w ≥ u) in Ω. Recently, Lu and Wang [14] found another proof of Theorem 2.14 using a different approa”
Inhomogeneous infinity Laplace equation (2008, unknown): “For an element S ∈ Sn×n, ‖S‖ denotes its operator norm, namely supx∈Rn\{0} 〈Sx,x〉 |x|2 . u ≺x0 ϕ means u − ϕ has a local maximum at x0.”