請用此 Handle URI 來引用此文件：
http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/80333完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 王振男(Jenn-Nan Wang) | |
| dc.contributor.author | Pei-Shan Fang | en |
| dc.contributor.author | 方佩珊 | zh_TW |
| dc.date.accessioned | 2022-11-24T03:04:39Z | - |
| dc.date.available | 2021-08-06 | |
| dc.date.available | 2022-11-24T03:04:39Z | - |
| dc.date.copyright | 2021-08-06 | |
| dc.date.issued | 2021 | |
| dc.date.submitted | 2021-07-06 | |
| dc.identifier.citation | [1] R. Adams. Sobolev spaces, acad. Press, New York, 19(5), 1975. [2] H. Berestycki, L. Nirenberg, and S. S. Varadhan. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Communications on Pure and Applied Mathematics, 47(1):47–92, 1994. [3] H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer Science Business Media, 2010. [4] D. G. de Figueiredo. Positive solutions of semilinear elliptic problems. In Differential equations, pages 34–87. Springer, 1982. [5] D. G. de Figueiredo and J.P. Gossezt. Strict monotonicity of eigenvalues and unique continuation. In Djairo G. de FigueiredoSelected Papers, pages 361–368. Springer, 1992. [6] E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional sobolev spaces. Bulletin des sciences mathématiques, 136(5):521–573, 2012. [7] M. Donsker and S. S. Varadhan. On the principal eigenvalue of secondorder elliptic differential operators. Communications on Pure and Applied Mathematics, 29(6):595–621, 1976. [8] Y. V. Egorov and V. A. Kondratiev. On spectral theory of elliptic operators, volume 89. Birkhäuser, 1996. [9] M. M. Fall and V. Felli. Unique continuation property and local asymptotics of solutions to fractional elliptic equations. Communications in Partial Differential Equations, 39(2):354–397, 2014. [10] S. Frassu and A. Iannizzotto. Strict monotonicity and unique continuation for general nonlocal eigenvalue problems. Taiwanese Journal of Mathematics, 24(3):681–694, 2020. [11] M.Á. GarcíaFerrero and A. Rüland. Strong unique continuation for the higher order fractional laplacian. arXiv preprint arXiv:1902.09851, 2019. [12] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, volume 224. springer, 1977. [13] D. S. Grebenkov and B.T. Nguyen. Geometrical structure of laplacian eigenfunctions. siam REVIEW, 55(4):601–667, 2013. [14] A. Henrot. Extremum problems for eigenvalues of elliptic operators. Springer Science Business Media, 2006. [15] P.Z. Kow and M. Kimura. The lewystampacchia inequality for fractional laplacian and its application to anomalous unidirectional diffusion equations. arXiv preprint arXiv:1909.00588, 2019. [16] P.Z. Kow and J.N. Wang. Strict monotinicity of eigenvalues and unique continuation for spectral fractional elliptic operators. [17] M. Kwaśnicki. Ten equivalent definitions of the fractional laplace operator. Fractional Calculus and Applied Analysis, 20(1):7–51, 2017. [18] J. Levandosky. Eigenvalues of the laplacian. [19] C. Martinez and M. Sanz. The theory of fractional powers of operators. Elsevier, 2001. [20] S. E. Mikhailov. Traces, extensions and conormal derivatives for elliptic systems on Lipschitz domains. Journal of mathematical analysis and applications, 378(1):324–342, 2011. [21] R. Musina and A. I. Nazarov. On fractional laplacians. Communications in Partial Differential Equations, 39(9):1780–1790, 2014. [22] R. D. Nussbaum and Y. Pinchover. On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. Journal d＇Analyse Mathématique, 59(1):161–177, 1992. [23] M. H. Protter. Unique continuation for elliptic equations. Transactions of the American Mathematical Society, 95(1):81–91, 1960. [24] M. H. Protter, H. F. Weinberger, et al. On the spectrum of general second order operators. Bulletin of the American Mathematical Society, 72(2):251–255, 1966. [25] A. Rüland. Unique continuation for fractional schrödinger equations with rough potentials. Communications in Partial Differential Equations, 40(1):77–114, 2015. [26] V. Serov. Fourier series, Fourier transform and their applications to mathematical physics, volume 197. Springer, 2017. [27] P. R. Stinga. User’s guide to the fractional laplacian and the method of semigroups. Fractional Differential Equations, pages 235–266, 2019. [28] P. R. Stinga and C. Zhang. Harnack’s inequality for fractional nonlocal equations. arXiv preprint arXiv:1203.1518, 2012. [29] H. Yu. Unique continuation for fractional orders of elliptic equations. Annals of PDE, 3(2):1–21, 2017. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/80333 | - |
| dc.description.abstract | 本篇論文以分數階橢圓算子在狄利克雷條件下的權函數(weight function)特徵值問題為出發點，探討在權函數固定的狀況下值域變動所得出之特徵值變化。對應於有界值域變動的狄利克雷條件拉普拉斯算子的特徵值單調性已被證明，我們將會證明分數階橢圓算子也有相同結果。此外，我們也會證明若將此特徵值問題的主要特徵值試為對值域的函數，則其為連續函數。 在證明上我們將會運用對應分數階橢圓算子的瑞利商(Rayleigh quotient)及一些相關的最大值—最小值定理。 | zh_TW |
| dc.description.provenance | Made available in DSpace on 2022-11-24T03:04:39Z (GMT). No. of bitstreams: 1 U0001-2206202101560900.pdf: 806436 bytes, checksum: 93f0d7fc2f9c8e2f9c21214aa3614a90 (MD5) Previous issue date: 2021 | en |
| dc.description.tableofcontents | "Verification Letter from the Oral Examination Committee i 摘要 iii Abstract v Contents vii Chapter 1 Introduction 1 Chapter 2 Some review on standard elliptic operator 3 2.1 Variational principles for the principal eigenvalue 4 2.2 Unique continuation property 5 Chapter 3 Fractional Sobolev space 11 3.1 Sobolev Spaces Wk,p(Ω) 11 3.2 Fractional Sobolev spaces 13 Chapter 4 The Spectral Fractional Operator L^γ 31 4.1 Definition of Spectral fractional Laplace operator 31 4.2 Definition of L^γ 33 Chapter 5 The weighted eigenvalue problem 37 5.1 The existence of eigenvalue 37 5.2 The monotonicity of eigenvalues depending on weight function 47 Chapter 6 The property of domain monotonicity 51 6.1 Rayleigh quotient of L^γ 51 6.2 The domain monotonicity of eigenvalues 57 6.3 The continuity of principal eigenvalue 60 Chapter 7 A counterexample for Neumann boundary condition 63 Appendix A — Equivalent definitions of the fractional Laplacian 67 A.1 Definitions of the fractional Laplacian 67 A.2 Difference from spectral fractional Laplacian 72 Appendix B — Unique continuation properties for fractional operators 75 References 79" | |
| dc.language.iso | zh-TW | |
| dc.subject | 狄利克雷邊界條件 | zh_TW |
| dc.subject | 分數型橢圓算子 | zh_TW |
| dc.subject | 分數型索伯列夫空間 | zh_TW |
| dc.subject | 特徵值問題 | zh_TW |
| dc.subject | 單調性 | zh_TW |
| dc.subject | 瑞利商 | zh_TW |
| dc.subject | fractional elliptic operator | en |
| dc.subject | monotonicity property | en |
| dc.subject | eigenvalue problem | en |
| dc.subject | fractional Sobolev space | en |
| dc.subject | Dirichlet boundary condition | en |
| dc.subject | Rayleigh quotient | en |
| dc.title | 譜分數型橢圓算子之特徵值單調性 | zh_TW |
| dc.title | Monotonicity of Eigenvalues of the Spectral Fractional Elliptic Operator | en |
| dc.date.schoolyear | 109-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 林景隆(Hsin-Tsai Liu),林奕亘(Chih-Yang Tseng) | |
| dc.subject.keyword | 分數型橢圓算子,分數型索伯列夫空間,特徵值問題,單調性,瑞利商,狄利克雷邊界條件, | zh_TW |
| dc.subject.keyword | fractional elliptic operator,fractional Sobolev space,eigenvalue problem,monotonicity property,Rayleigh quotient,Dirichlet boundary condition, | en |
| dc.relation.page | 82 | |
| dc.identifier.doi | 10.6342/NTU202101088 | |
| dc.rights.note | 同意授權(限校園內公開) | |
| dc.date.accepted | 2021-07-07 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
文件中的檔案：
| 檔案 | 大小 | 格式 | |
|---|---|---|---|
| U0001-2206202101560900.pdf 授權僅限NTU校內IP使用（校園外請利用VPN校外連線服務） | 787.54 kB | Adobe PDF |
系統中的文件，除了特別指名其著作權條款之外，均受到著作權保護，並且保留所有的權利。