The boundary Harnack inequality for infinity harmonic functions in the plane. (English summary) Proc. Amer. Math. Soc. 136 (2008), no. 4, 1311–1323. In this paper the authors prove a boundary Harnack inequality for positive ∞-harmonic functions (∆∞u = 0) vanishing on a portion of the boundary of some special type domains. More precisely, Ω ⊂ R 2 is a bounded domain with a geometric assumption which is equivalent to assuming that ∂Ω is a “quasicircle”. With respect to the proof, the ∞-Laplacian (∆∞) can often be understood by considering limits on the corresponding problem to the p-Laplacian (∆p), for 1 < p < ∞. In fact, the proof of the boundary Harnack inequality is based on this approach; namely, it is based on a uniform, with respect to p, boundary Harnack inequality for large values of p
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