臺灣博碩士論文加值系統

本論文文中，我們主要是研讀Talenti 所寫的 "Best Constant in Sobolev Inequality"
athored 和Agueh 所寫的 "A New ODE Approach To Sharp Sobolev Inequalities" 這兩篇
論文，並且研究這兩個工作的細節和比較這兩個關於索伯列夫不等式最佳常數逼近法的
差異。
在Talenti 的論文中，Talenti 的證明由兩個部分組成。第一步利用了球對稱重排去
簡化問題，找出問題所對應極致曲線及其對應的最佳函數，第二步Talenti 證明了在第
一部分給出的極致曲線的確給了我們最大值並找出索伯利夫不等式的最佳常數。
在Agueh 的論文中，他藉由研究Talenti 和 另一個由Cordero-Erausquin 提出的逼近
法之間的聯繫，找到了個變數變換幫助我們把從Talenti 獲得的極致曲線簡化成一個非
線性的常微分方程解出並由此獲得索伯列夫不等式最佳常數。

In this thesis, we mainly study the paper with title "Best Constant in Sobolev Inequality"
authored by Talenti and "A New ODE Approach To Sharp Sobolev Inequalities" authored by
Agueh, elaborate the detail of the work and compare these two approaches of best constant in
Sobolev Inequality.
In Talenti's approaches, Talenti proves the best constant for the Sobolev Inequality and
the proof consists with twos step. In the first step, talents proofs are based on a spherically
symmetric function which helped them reduce the problem, and find the extremal function
and its solution. In the second step, Talenti shows that the extremal found in step 1 actually
gives the maximum and finds the best constant in Sobolev Inequality.
In Agueh's approaches Agueh investigates the link between Talenti's approaches and
another approach proposed by Cordero-Erausquin. We show that a strategic change of
variable which helps to solve explicitly the nonlinear ordinary differential equation -- leading
to the sharp constant and extremals in Sobolev Inequality obtained from the approach by
Talenti.

1 Introduction
1.1 Background
1.2 Sobolev Inequality
1.3 Talenti’s Approach
1.4 A ODE Approach .
2 Preknowledge
2.1 Lp space
2.2 Elementary Inequalities
2.3 Weak Derivatives
2.4 Wk;p Spaces
2.5 Lemmas and De…nitions
2.6 Proof of Sobolev Inequaility
3 Tanlti’s Approach[1976]
3.1 Spherically symmetric functions
3.2 Extremal of J(u)
3.3 Lagrange problem .
4 Agueh’s ODE Approach
4.1 Mass Transportation Approach
4.2 An Optimization Problem .
4.3 Sharp Sobolev Inequalities

[1] Talenti, Giorgio. "Best constant in Sobolev inequality." Annali di Matem-
atica pura ed Applicata 110.1 (1976): 353-372.
[2] T. Aubin. (1976). Probleme isoperimetrique et espaces de Sobolev, J.
Di¤erential Geometry.11, 573-598.
[3] Agueh, M. "A new ODE approach to sharp Sobolev inequalities." Non-
linear analysis research trends (2008): 1-13.
[4] Bliss, G. A. "An integral inequality." Journal of the LondonMathematical
Society 1.1 (1930): 40-46.
[5] Cordero-Erausquin, Dario, Bruno Nazaret, and Ceric Villani. "A mass-
transportation approach to sharp Sobolev and Gagliardo–Nirenberg in-
equalities." Advances in Mathematics 182.2 (2004): 307-332.
[6] Lieb, Elliott H. "Sharp constants in the Hardy-Littlewood-Sobolev and
related inequalities." Inequalities. Springer, Berlin, Heidelberg, 2002. 529-
554.
[7] McCann, Robert John. A convexity theory for interacting gases and equi-
librium crystals. Diss. Princeton University, 1994.
[8] Agueh, Martial. "Sharp Gagliardo–Nirenberg inequalities via p-Laplacian
type equations." Nonlinear Di¤erential Equations and Applications
NoDEA 15.4-5 (2008): 457-472.
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