On the curvature of level sets of harmonic functions

2013/07/08 by Steinerberger, Stefan
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1307.2069

Abstract

If a real harmonic function inside the open unit disk B(0,1) ⊂ ℝ² has its level set \x: u(x) = u(0)\ diffeomorphic to an interval, then we prove the sharp bound κ≤ 8 on the curvature of the level set \x: u(x) = u(0)\ in the origin. The bound is sharp and we give the unique (up to symmetries) extremizer.

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