## Abstract

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 sup_{K}≤|f|≤Cinf_{K}≤|f|andsup_{K}≤|∇;f|≤Cinf_{K}≤|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).

Original language | English (US) |
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Pages (from-to) | 241-262 |

Number of pages | 22 |

Journal | Advances in Mathematics |

Volume | 274 |

DOIs | |

State | Published - Apr 9 2015 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- 31B05
- 35B09
- 35B50
- Boundary Harnack principle
- Gradient estimates
- Harmonic functions
- Harmonic polynomials
- Harnack inequality
- Maximum principle
- Nodal set