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標題Title: NODAL SETS OF LAPLACE EIGENFUNCTIONS: POLYNOMIAL UPPER ESTIMATES OF THE HAUSDORFF MEASURE.
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: Riemannian manifold
摘要Abstract: Let M be a compact C∞-smooth Riemannian manifold of dimension n, n ≥ 3, and let φλ : ∆Mφλ + λφλ = 0 denote the Laplace eigenfunction on M corresponding to the eigenvalue λ. We show that
Hn−1({φλ = 0}) ≤ Cλα,
where α > 1/2 is a constant, which depends on n only, and C > 0 depends on M . This result is a consequence of our study of zero sets of harmonic functions on C∞-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.

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2016_6_8643f2a1.pdf 203Kb pdf 33 4
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