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| 研究生： |
高子耘 Gao, Zi-Yun |
|---|---|
| 論文名稱： |
流形上的p-拉普拉斯算子研究 A Study of p-Laplacian on Complete Manifolds |
| 指導教授： |
宋瓊珠
Sung, Chiung-Jue |
| 口試委員: |
高淑蓉
Kao, Shu-Jung 饒維明 Nhieu, Duy-Minh |
| 學位類別： |
碩士 Master |
| 系所名稱： |
理學院 - 數學系 Department of Mathematics |
| 論文出版年： | 2022 |
| 畢業學年度： | 110 |
| 語文別： | 英文 |
| 論文頁數： | 60 |
| 中文關鍵詞： | 流形 、拉普拉斯 、梯度估計 |
| 外文關鍵詞： | manifold, Lalplacian, gradient estimate |
| 相關次數： | 點閱：1 下載：0 |
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在本論文中，我們首先給出了拉普拉斯特徵函數梯度估計的完整證明，然後給出了p-特徵函數的銳梯度估計。最後，我們詳細證明了主特徵值達到最大值的流形結構。
In this thesis, we first give a complete proof of a gradient estimate for positive eigenfunctions of Laplacian, then we show a sharp gradient estimate for positive p-eigenfunctions. At last, we give a detailed proof of the theorem of Sung and Wang in the structure of manifolds whose principal eigenvalues achieve the maximum value.
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[2] P. Li and J. Wang, Complete manifolds with positive spectrum, II. Journal of Differential Geometry 62.1 (2002), 143-162.
[3] X. Wang and L. Zhang, Local gradient estimate for p-harmonic functions on Riemannian manifolds, Communications in Analysis and Geometry 19 (2011), 759–772.
[4] C. Sung and J. Wang, Sharp gradient estimate and spectral rigidity for p -Laplacian, Mathematical Research Letters. 21 (2014), 885-904
[5] P. Li, Geometric analysis. Vol. 134. Cambridge University Press, (2012).
[6] L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds. Journal of Differential Geometry, 36.2 (1992), 417-450.
[7] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of non-negative Ricci curvature. Journal of Differential Geometry, 6.1 (1971),
119- 128.