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Reviewed by Eduardo Colorado References

By John L. (-ky) Nyström and G. Aronsson


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

Topics: 2. G. Aronsson, Minimization problems for the functional supx F (x, f (x, f (x)) II, Arkiv f”or
Year: 2013
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