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標題Title: NODAL SETS OF LAPLACE EIGENFUNCTIONS: PROOF OF NADIRASHVILI’S CONJECTURE AND OF THE LOWER BOUND IN YAU’S CONJECTURE.
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: harmonic function
摘要Abstract: Let u be a harmonic function in the unit ball B(0, 1) ⊂ Rn, n ≥ 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hn−1({u = 0} ∩ B) ≥ c. We prove Nadirashvili’s conjecture as well as its counterpart on C∞-smooth Riemannian manifolds. The latter yields the lower bound in Yau’s conjecture. Namely, we show that for any compact C∞-smooth Riemannian manifold M (without boundary) of dimension n there exists c > 0 such that for any Laplace eigenfunction φλ on M, which corresponds to the eigenvalue λ, the following inequality holds: c√λ ≤ Hn−1({φλ = 0}).

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2016_6_92558438.pdf 206Kb pdf 31 2
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