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標題Title: RATIOS OF HARMONIC FUNCTIONS WITH THE SAME ZERO SET
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: harmonic functions
摘要Abstract: We study the ratio of harmonic functions u, v which have thesamezerosetZintheunitballB⊂Rn. Theratiof=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 in any dimension: for a given compact set K ⊂ B we show that supK |f| ≤ C1 infK |f| and supK |∇f| ≤ C2 infK |f|, where C1 and C2 depend on K and Z only. In dimension two we specify the dependence of the constants on Z in these inequalities by showing that only the number of nodal domains of u, i.e. the number of connected components of B\Z, plays a role.

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2016_6_63c92f03.pdf 222Kb pdf 70 1
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