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標題Title: ON RATIOS OF HARMONIC FUNCTIONS
作者Authors: ALEXANDER LOGUNOV AND EUGENIA MALINNIKOVA
上傳單位Department: 電子工程系
上傳時間Date: 2016-6-8
上傳者Author: 傅俊結
審核單位Department: 電子工程系
審核老師Teacher: 傅俊結
檔案類型Categories: 論文Thesis
關鍵詞Keyword: maximum and minimum principles
摘要Abstract: Let u and v be harmonic in Ω ⊂ Rn functions 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|f| ≤ C inf |f| & sup|∇f| ≤ C inf |f| for any compact subset K KKKK
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).

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2016_6_8689b9ce.pdf 207Kb pdf 30 1
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