Electronic Research Announcements in Mathematical Sciences (ERA-MS)

A gradient estimate for harmonic functions sharing the same zeros

Pages: 62 - 71, Volume 21, 2014      doi:10.3934/era.2014.21.62

       Abstract        References        Full Text (358.7K)              Related Articles       

Dan Mangoubi - Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel (email)

Abstract: Let $u, v$ be two harmonic functions in $\{|z|<2\}\subset\mathbb{C}$ which have exactly the same set $Z$ of zeros. We observe that $\big|\nabla\log |u/v|\big|$ is bounded in the unit disk by a constant which depends on $Z$ only. In case $Z=\emptyset$ this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives only Hölder estimates on $\log |u/v|$.

Keywords:  Li-Yau, harmonic functions, nodal set, gradient estimates, Harnack, boundary Harnack principle.
Mathematics Subject Classification:  31B05, 35J15.

Received: June 2013;      Revised: December 2013;      Available Online: May 2014.