TY - JOUR

T1 - On ratios of harmonic functions

AU - Logunov, A.

AU - Malinnikova, E.

PY - 2015

Y1 - 2015

N2 - © 2015 Elsevier Inc.Let u and v be harmonic functions in Ω⊂Rn with the same zero set Z. We show that the ratio f of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum principles. For n=3 we also prove the Harnack inequality and the gradient estimate for the ratios of harmonic functions, namely supK≤|f|≤CinfK≤|f|andsupK≤|∇;f|≤CinfK≤|f| for any compact subset K of Ω, where the constant C depends on K, Z, Ω only. In dimension two the first inequality follows from the boundary Harnack principle and the second from the gradient estimate recently obtained by Mangoubi. It is an open question whether these inequalities remain true in higher dimensions (n≥4).

AB - © 2015 Elsevier Inc.Let u and v be harmonic functions in Ω⊂Rn with the same zero set Z. We show that the ratio f of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum principles. For n=3 we also prove the Harnack inequality and the gradient estimate for the ratios of harmonic functions, namely supK≤|f|≤CinfK≤|f|andsupK≤|∇;f|≤CinfK≤|f| for any compact subset K of Ω, where the constant C depends on K, Z, Ω only. In dimension two the first inequality follows from the boundary Harnack principle and the second from the gradient estimate recently obtained by Mangoubi. It is an open question whether these inequalities remain true in higher dimensions (n≥4).

U2 - 10.1016/j.aim.2015.01.009

DO - 10.1016/j.aim.2015.01.009

M3 - Article

SP - 241

EP - 262

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -