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On ratios of harmonic functions II | | Alexander Logunov
; Eugenia Malinnikova
; | | Date: |
26 Jun 2015 | | Abstract: | We study the ratio of harmonic functions $u,v$ which have the same zero set
$Z$ in the unit ball $Bsubset mathbb{R}^n$. The ratio $f=u/v$ can be extended
to a real analytic, nowhere vanishing function in $B$. We prove the Harnack
inequality and the gradient estimate for such ratios: for a given compact set
$Ksubset B$ we show that $sup_K|f|le C_1inf_K|f|$ and $sup_Kleft|
abla
f
ight|le C_2 inf_K|f|$, where $C_1$ and $C_2$ depend only on $K,Z$. In
dimension two we specify these estimates by showing that only the number of
nodal domains of $u$ plays a role. | | Source: | arXiv, 1506.8041 | | Services: | Forum | Review | PDF | Favorites | |
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