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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 李瑩英(Yng-Ing Lee) | |
| dc.contributor.author | Yuan Shyong Ooi | en |
| dc.contributor.author | 黃垣熊 | zh_TW |
| dc.date.accessioned | 2021-06-16T06:44:20Z | - |
| dc.date.available | 2020-08-04 | |
| dc.date.copyright | 2020-08-04 | |
| dc.date.issued | 2020 | |
| dc.date.submitted | 2020-07-28 | |
| dc.identifier.citation | [All75] William K Allard. On the first variation of a varifold: boundary behavior. Annals of Mathematics, pages 418–446, 1975. [Alm84] F Almgren. Q valued functions minimizing dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two. Princeton, NJ, 1984. [BDGG69] E. Bombieri, E. De Giorgi, and E. Giusti. Minimal cones and the Bernstein problem. Invent. Math., 7:243–268, 1969. [CCH12] Albert Chau, Jingyi Chen, and Weiyong He. Lagrangian mean curvature flow for entire Lipschitz graphs. Calc. Var. Partial Differential Equations, 44(1-2): 199–220, 2012. [Che79] Jeff Cheeger. On the spectral geometry of spaces with cone-like singularities. Proc. Nat. Acad. Sci. U.S.A., 76(5):2103–2106, 1979. [Che83] Jeff Cheeger. Spectral geometry of singular Riemannian spaces. J. Differential Geom., 18(4):575–657 (1984), 1983. [CHS84] Luis Caffarelli, Robert Hardt, and Leon Simon. Minimal surfaces with isolated singularities. Manuscripta Math., 48(1-3):1–18, 1984. [CWY09] Jingyi Chen, Micah Warren, and Yu Yuan. A priori estimate for convex solutions to special Lagrangian equations and its application. Comm. Pure Appl. Math., 62(4):583–595, 2009. [DJX16] Qi Ding, Jürgen Jost, and Yuanlong Xin. Minimal graphic functions on manifolds of nonnegative Ricci curvature. Comm. Pure Appl. Math., 69(2):323– 371, 2016. [DJX17] Qi Ding, J Jost, and YL Xin. Existence and non-existence of minimal graphs. arXiv preprint arXiv:1701.01674, 2017. [Fan51] Ky Fan. Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Nat. Acad. Sci., U. S. A., 37:760–766, 1951. [FC80] D. Fischer-Colbrie. Some rigidity theorems for minimal submanifolds of the sphere. Acta Math., 145(1-2):29–46, 1980. [GM13] Mariano Giaquinta and Luca Martinazzi. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Springer Science Business Media, 2013. [Gre94] Giovanni Gregori. Compactness and gradient bounds for solutions of the mean curvature system in two independent variables. J. Geom. Anal., 4(3): 327–360, 1994. [GT01] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. [HL82] Reese Harvey and H. Blaine Lawson, Jr. Calibrated geometries. Acta Math., 148:47–157, 1982. [HS85] Robert Hardt and Leon Simon. Area minimizing hypersurfaces with isolated singularities. Journal für die reine und angewandte Mathematik, 1985(362): 102–129, 1985. [JS68] Howard Jenkins and James Serrin. The dirichlet problem for the minimal surface equation in higher dimensions. J. reine angew. Math, 229:170–187, 1968. [JXY15] J. Jost, Y. L. Xin, and Ling Yang. Curvature estimates for minimal submanifolds of higher codimension and small G-rank. Trans. Amer. Math. Soc., 367(12):8301–8323, 2015. [JXY16] Jürgen Jost, Yuan Long Xin, and Ling Yang. The geometry of Grassmannian manifolds and Bernstein-type theorems for higher codimension. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 16(1):1–39, 2016. [JXY18] Jürgen Jost, Yuanlong Xin, and Ling Yang. Submanifolds with constant Jordan angles and rigidity of the Lawson-Osserman cone. Asian J. Math., 22(1):75–109, 2018. [Li20] Yang Li. Dirichlet problem for maximal graphs of higher codimension. Int. Math. Res. Not. IMRN, 2020. rnaa099. [LO77] H. B. Lawson, Jr. and R. Osserman. Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta Math., 139(1-2): 1–17, 1977. [LOT19] Yng-Ing Lee, Yuan Shyong Ooi, and Mao-Pei Tsui. Uniqueness of minimal graph in general codimension. J. Geom. Anal., 29(1):121–133, 2019. [LT14] Yng-Ing Lee and Mao-Pei Tsui. Stability of the minimal surface system and convexity of area functional. Trans. Amer. Math. Soc., 366(7):3357–3371, 2014. [LW03] Yng-Ing Lee and Mu-Tao Wang. A stability criterion for nonparametric minimal submanifolds. Manuscripta Math., 112(2):161–169, 2003. [LW08] Yng-Ing Lee and Mu-Tao Wang. A note on the stability and uniqueness for solutions to the minimal surface system. Math. Res. Lett., 15(1):197–206, 2008. [Mir59] L. Mirsky. On a convex set of matrices. Arch. Math., 10:88–92, 1959. [MJ53] Charles B Morrey Jr. Second order elliptic systems of differential equations. Proceedings of the National Academy of Sciences of the United States of America, 39(3):201, 1953. [MJ09] Charles Bradfield Morrey Jr. Multiple integrals in the calculus of variations. Springer Science Business Media, 2009. [PR00] Frank Pacard and Tristan Rivière. Linear and nonlinear aspects of vortices, volume 39 of Progress in Nonlinear Differential Equations and their Appli- cations. Birkhäuser Boston, Inc., Boston, MA, 2000. The Ginzburg-Landau model. [Rad30] Tibor Radó. On plateau’s problem. Annals of Mathematics, pages 457–469, 1930. [Sim68] James Simons. Minimal varieties in riemannian manifolds. Ann. of Math. (2), 88:62–105, 1968. [Sim83] Leon Simon. Lectures on geometric measure theory, volume 3 of Proceed- ings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983. [Sma87] Nathan Smale. A bridge principle for minimal and constant mean curvature submanifolds of R^N . Invent. Math., 90(3):505–549, 1987. [Sma89] Nathan Smale. Minimal hypersurfaces with many isolated singularities. Annals of Mathematics, 130(3):603–642, 1989. [Sma91] Nathan Smale. An equivariant construction of minimal surfaces with non-trivial singular sets. Indiana Univ. Math. J., 40(2):595–616, 1991. [Sma93] Nathan Smale. Geometric P.D.E.s with isolated singularities. J. Reine Angew. Math., 440:1–41, 1993. [Wan03] Mu-Tao Wang. On graphic Bernstein type results in higher codimension. Trans. Amer. Math. Soc., 355(1):265–271, 2003. [Wan04a] Mu-Tao Wang. The dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions. Communications on Pure and Applied Mathematics, 57(2):267–281, 2004. [Wan04b] Mu-Tao Wang. Interior gradient bounds for solutions to the minimal surface system. Amer. J. Math., 126(4):921–934, 2004. [WY08] Micah Warren and Yu Yuan. A Liouville type theorem for special Lagrangian equations with constraints. Comm. Partial Differential Equations, 33(4-6): 922–932, 2008. [WY09] Micah Warren and Yu Yuan. Explicit gradient estimates for minimal Lagrangian surfaces of dimension two. Math. Z., 262(4):867–879, 2009. [Xin19] Yuanlong Xin. Minimal submanifolds and related topics, volume 16 of Nankai Tracts in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. Second edition of [ MR2035469]. [XYZ19] Xiaowei Xu, Ling Yang, and Yongsheng Zhang. Dirichlet boundary values on euclidean balls with infinitely many solutions for the minimal surface system. Journal de Mathématiques Pures et Appliquées, 129:266–300, 2019. [Zha18] Yongsheng Zhang. On non-existence of solutions of the dirichlet problem for the minimal surface system. arXiv preprint arXiv:1812.11553, 2018. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/57394 | - |
| dc.description.abstract | 高餘維極小子流形有很多性質是不同於超曲面的情況。文章中我們主要探討高餘維極小子流形的存在性和唯一性問題。 論文前半部將探討圖形式 (graphical) 極小子流形的唯一性問題。圖形式極小子流形滿足的微分方程稱之為極小曲面方程 (minimal surface system). 根據 [LO77], 極小曲面方程的狄利克雷問題 (Dirichlet problem) 的解一般並不會有唯一性。我們通過凸優化理論的工具，證明了當狄利克雷問題的解的奇異值 (singular value) 滿足特定條件時，它就會有唯一性。 論文後半部討論有奇異點的極小子流形的存在性問題。我們推廣了 [CHS84] 關於帶奇異點極小超曲面的結果到高餘維度的情況。我們證明了給定正則的極小錐，可以在它附近找到帶奇異點非錐的高餘維極小子流形。 | zh_TW |
| dc.description.abstract | Higher codimension minimal submanifold behaves differently from the hypersurface case. In this thesis, we focus on the aspect of existence and uniqueness of higher codimension minimal submanifold. The first part of this thesis discusses the uniqueness property of graphical minimal submanifold. The uniqueness of the Dirichlet problem of minimal surface system is generally false as noted by [LO77]. By using tools from convex optimization theory, we are able to give sufficient condition in term of singular value for the uniqueness problem. The second part discusses the existence of general minimal submanifold with isolated singularity in higher codimension but is not a cone. We generalize the Leray fix-point method in [CHS84] to higher codimension setting and show the existence of non-conical higher codimension minimal submanifold (with boundary) with isolated singularity. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T06:44:20Z (GMT). No. of bitstreams: 1 U0001-2107202015275700.pdf: 1501543 bytes, checksum: f528015c1d326b4600be7618c93e6554 (MD5) Previous issue date: 2020 | en |
| dc.description.tableofcontents | 口試委員會審定書 i 致謝 ii 中文摘要 iii Abstract iv Contents v 1 Introduction 1 1.1 Graphical Minimal Submanifold . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Minimal Surface System . . . . . . . . . . . . . . . . . . . . . . 1 1.2 General Minimal submanifold . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Minimal Submanifold with Isolated singulaties . . . . . . . . . .9 2 Background Materials 15 2.1 Submanifold Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Second fundamental form and mean curvature . . . . . . . . . .15 2.1.2 Second variational formula for area . . . . . . . . . . . . . . . . 20 2.2 Some Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Singular Value Decomposition . . . . . . . . . . . . . . . . . . .22 2.2.2 Weak Majorization . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Weighted Holder Space in $R^n$ . . . . . . . . . . . . . . . . . . . . . . . .24 3 Uniqueness of Dirichlet problem for minimal surface system 27 3.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 3.2 Application of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . .30 3.3 Uniqueness results in general Riemannian manifold . . . . . . . . . . . .34 4 Minimal submanifold with isolated singularity 36 4.1 $L^2$ solution of linearized equation . . . . . . . . . . . . . . . . . . . . . 37 4.2 Weighted Holder estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.1 Weighted Holder space on $C^*_1$ . . . . . . . . . . . . . . . . . . . 42 4.2.2 Weighted Schauder estimate . . . . . . . . . . . . . . . . . . . . 44 4.2.3 weighted $C^0$ estimate . . . . . . . . . . . . . . . . . . . . . . . .57 4.3 Solving nonlinear equation . . . . . . . . . . . . . . . . . . . . . . . . .65 Bibliography 71 | |
| dc.language.iso | en | |
| dc.title | 高餘維子流形的存在性和唯一性 | zh_TW |
| dc.title | Existence and Uniqueness of Minimal Submanifold in higher codimension | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 108-2 | |
| dc.description.degree | 博士 | |
| dc.contributor.oralexamcommittee | 崔茂培(Mao-Pei Tsui),張樹城(Shu-Cheng Chang),蔡忠潤(Chung-Jun Tsai),林俊吉(Chun-Chi Lin),蔡東和(Dong-Ho Tsai) | |
| dc.subject.keyword | 極小子流形,高餘維,極小曲面方程,狄利克雷問題,奇異值,孤立奇異點, | zh_TW |
| dc.subject.keyword | minimal submanifold,higher codimension,minimal surface system,Dirichlet problem,singular value,isolated singularity, | en |
| dc.relation.page | 75 | |
| dc.identifier.doi | 10.6342/NTU202001694 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2020-07-29 | |
| 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 | |
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