A gradient estimate for harmonic functions sharing the same zeros

Dan Mangoubi
2014 Electronic Research Announcements in Mathematical Sciences  
Let u, v be two harmonic functions in the disk of radius two which have exactly the same set Z of zeros. We observe that the gradient of \log |u/v| is bounded in the unit disk by a constant which depends on Z only. In case Z is empty this goes back to Li-Yau's gradient estimate for positive harmonic functions. The general boundary Harnack principle gives H\"older estimates on \log |u/v|.
doi:10.3934/era.2014.21.62 fatcat:2dpbr3porveenaewtt7jox4hda