On the curvature of level sets of harmonic functions

Stefan Steinerberger

On the curvature of level sets of harmonic functions

Classical Analysis and ODEs 2014-07-02 v5 Analysis of PDEs

Authors: Stefan Steinerberger

Abstract

If a real harmonic function inside the open unit disk $B(0,1) \subset \mathbb{R}^2$ has its level set $\left\{x: u(x) = u(0)\right\}$ diffeomorphic to an interval, then we prove the sharp bound $\kappa \leq 8$ on the curvature of the level set $\left\{x: u(x) = u(0)\right\}$ in the origin. The bound is sharp and we give the unique (up to symmetries) extremizer.

Keywords

harmonic functions level sets curvature unit disk sharp bound extremizer

Cite

@article{arxiv.1307.2069,
  title  = {On the curvature of level sets of harmonic functions},
  author = {Stefan Steinerberger},
  journal= {arXiv preprint arXiv:1307.2069},
  year   = {2014}
}

Comments

This paper is withdrawn: the result has been proven earlier by Kuran (On the zeros of harmonic functions, J. London Math. Soc. 44, 1969, 303-309)

On the curvature of level sets of harmonic functions

Abstract

Keywords

Cite

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