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http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/65733完整後設資料紀錄
| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 王藹農(Ai-Nung Wang) | |
| dc.contributor.author | Wei-Ming Hsu | en |
| dc.contributor.author | 許為明 | zh_TW |
| dc.date.accessioned | 2021-06-17T00:02:17Z | - |
| dc.date.available | 2022-12-31 | |
| dc.date.copyright | 2012-07-27 | |
| dc.date.issued | 2012 | |
| dc.date.submitted | 2012-07-16 | |
| dc.identifier.citation | [1] J. B. Conway, A course in functional analysis, Second edition, Springer, New York, 2007.
[2] D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal. 161, 257-269, 2002. [3] D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Advances in Mathematics, 182, pp. 307-332, 2004. [4] M. D. Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and application to nonlinear diffusions, J. Math. Pures Appl. 81 (9) 847-875, 2002. [5] M. D. Pino and J. Dolbeault, The optimal Euclidean L p -Sobolev logarithmic inequality, 2000. [6] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, Vol.19, Amer. Math. Soc., 1998. [7] L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current development in mathematics, Int. Press, Boston, 65-126, 1999. [8] L. C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, 1992. [9] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 653, 1999. [10] G. B. Folland, Real analysis: modern techniques and their applications, second edition, A John Wiley and Sons, Inc., publication, 1999. [11] W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math. 177, 113-161, 1996. [12] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Second edition, Springer, Berlin,2001. [13] P. D. Lax, Functional analysis, A John Wiley and Sons, Inc., publication, 2002. [14] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 1996. [15] R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (2), 309-323, 1995. [16] R. J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1), 153-179, 1997. [17] N. S. Trudinger and X.-J. Wang, On the Monge mass transfer problem, Calc. Var. PDE, 13, 19-31, 2001. [18] P. R. Thie, An Introduction to Linear Programming and Game Theory, John Wiley and Sons, 1979. [19] G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. (IV) 110, 353-372, 1976. [20] C. Villani, Topics in mass transportation, Graduate Studies in Mathematics, Vol. 58, Amer. Math. Soc., Providence, RI, 2003. [21] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (6), 1182-1238, 1978. [22] R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Monographs and textbooks in pure and applied mathematics 43, Taylor and Francis Group, LLC, CRC Press, 1977. [23] W. P. Ziemer, Weakly differentialble functions: Sobolev space and functions of bounded variation, Springer, Berlin, 1989. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/65733 | - |
| dc.description.abstract | 在這篇論文當中，我們首先證明在特殊的二次成本函數之下的蒙日質量運輸問題是有解的，而且我們將使用在此問題中所構造出來的最優映射來幫助我們去確定某些索伯列夫型不等式中的最佳常數。在此我們呈現出質量運輸方法提供一個很基本的方式去研究某些索伯列夫型不等式。我們在在n維歐氏空間中使用此方法時並未用到其歐氏結構。為了完成這次的工作，我們的主要參考文獻為[3]和[7]。 | zh_TW |
| dc.description.abstract | In this thesis, we first give a proof of the Monge mass transport problem for the special quadratic cost function, and use the optimal map which we constructed to sharp certain Sobolev-type inequalities. We show that mass transportation methods provide an elementary approach to the study of certain Sobolev-type inequalities. The Euclidean structure of n-dimensional Euclidean space plays no role in our approach. Besides, to complete our work, we
mainly consult the paper [3] and [7]. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T00:02:17Z (GMT). No. of bitstreams: 1 ntu-101-R99221010-1.pdf: 548385 bytes, checksum: e592a90bd2372e074701f6ce2de83903 (MD5) Previous issue date: 2012 | en |
| dc.description.tableofcontents | 謝辭………………………………………………………………………………... i
中文摘要………………………………………………………………….….…… ii 英文摘要……………………………………………………………………….…. iii 1 Introduction……………………………………………………………………. 1 2 Preliminary ……………………………………………………………………. 5 2.1 Mass transport……………………….…………………………….…….. 5 2.2 Legendre transform………...…………………...……………………….. 8 2.3 Arzela-Ascoli Theorem …….……………………………………..….…. 8 2.4 Regularization and approximation by mollification ………..…................ 9 2.5 Weak derivatives and Sobolev spaces ………………………………..... 10 2.6 BV functions …………………………………………………….….….. 12 2.7 Second derivatives a.e. for convex functions …………………….……. 13 2.8 Solving the Monge-Ampere equation ………………………………….. 15 2.9 The isoperimetric inequality ……………………………………..…..… 16 3 Monge mass transport problem ………………………………….………....... 17 3.1 Motivation from linear programming ……………………………….…… 17 3.2 Kantorovich dual problem ………………….…………………….….…... 17 3.3 Construction of an optimal map to the Monge problem……………..…… 28 4 Sharp Sobolev inequalities…………………………………………………… 33 4.1 Introduction and some important tools …………………………………... 33 4.2 Sharp Sobolev inequalities for the case p > 1 ……………………….…… 41 4.3 Sharp Sobolev inequalities for the case p = 1 ……………………………. 50 4.4 Equality cases of Sobolev inequalities…………………………………… 51 5 Sharp Galiardo-Nirenberg inequalities……………………………………….. 67 5.1 Galiardo-Nirenberg inequalities ………………….……………….……... 67 5.2 Equality cases of Proposition 5.1…………………….…………………... 78 Reference………………….………………………………………………….…... 79 | |
| dc.language.iso | en | |
| dc.subject | 加里亞爾多-尼倫堡不等式 | zh_TW |
| dc.subject | 最佳常數索伯列夫不等式 | zh_TW |
| dc.subject | 蒙日質量傳遞問題 | zh_TW |
| dc.subject | 最優運輸 | zh_TW |
| dc.subject | Gagliardo-Nirenberg inequality | en |
| dc.subject | Monge mass transport problem | en |
| dc.subject | sharp constants | en |
| dc.subject | Sobolev inequality | en |
| dc.subject | optimal transport | en |
| dc.title | 以最優運輸方法確定某些索伯列夫型不等式中的最佳常數 | zh_TW |
| dc.title | Sharp certain Sobolev-type inequalities via optimal transport | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 100-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 陳俊全(Chiun-Chuan Chen),陳界山(Jein-Shan Chen) | |
| dc.subject.keyword | 最優運輸,蒙日質量傳遞問題,最佳常數索伯列夫不等式,加里亞爾多-尼倫堡不等式, | zh_TW |
| dc.subject.keyword | optimal transport,Monge mass transport problem,sharp constants,Sobolev inequality,Gagliardo-Nirenberg inequality, | en |
| dc.relation.page | 81 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2012-07-16 | |
| 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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