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標題Title: ON THE BOUNDARY BEHAVIOR OF POSITIVE SOLUTIONS OF ELLIPTIC DIFFERENTIAL EQUATIONS
作者Authors: ALEXANDER LOGUNOV
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: positive harmonic function
摘要Abstract: Let u be a positive harmonic function in the unit ball B1 ⊂ Rn and let μ be the boundary measure of u. Consider a point x ∈ ∂B1 and let n(x) denote the unit normal vector at x. Let α be a numberin(−1,n−1]andA∈[0,+∞). Weprovethatu(x+n(x)t)tα →Aast→+0ifandonlyif
μ(Br(x))rα → CαA as r → +0, where Cα = πn/2 . For α = 0 it follows from the theorems rn−1 Γ( n−α+1 )Γ( α+1 )
22
by Rudin and Loomis which claim that a positive harmonic function has a limit along the normal iff the boundary measure has the derivative at the corresponding point of the boundary. For α = n − 1 it concerns about the point mass of μ at x and it follows from the Beurling minimal principle. For the general case of α ∈ (−1, n − 1) we prove it with the help of the Wiener Tauberian theorem in a similar way to Rudin’s approach.

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