加權等周不等式之ABP方法及其應用

Weighted Isoperimetric Inequalities via the ABP method and its Applications

林冠宇(Kuan-Yu Lin)

指導教授 ： 林長壽

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.10032

摘要

在這篇論文中，我們探討加權之等周不等式。對於所有固定加權體積之可測集，
我們的目標是刻劃使加權周長最小的所有可能。對於所有具有特定” 凹特性” 之權
重，透過ABP 方法，所有此種” 等周集” 可被完整地刻劃。
特別地，將此定理運用至某些” 單項式權重”，我們可以證明具有此種權重之
Sobolev、Trudinger、以及Morrey 不等式。

關鍵字

等周不等式 ； ABP方法 ； Neumann問題 ； Sobolev不等式 ； 單項式權重.

並列摘要

In this thesis, we study isoperimetric problems with weights following [Cabre
et al., 2013]. Given a positive function $w$ on $mathbb{R}^n$ (called a weight), our goal is to characterize minimizers of the weighted perimeter $int_{partial E} w,mathrm{d}S$ among all measurable sets E with a fixed weighted volume
$int_{E} w , mathrm{d}x$.
The result applies to all homogeneous weights satisfying certain concavity conditions, and the proof is achieved by applying the ABP method to an appropriate linear Neumann problem.
In particular, by applying this result to the monomial weight $|x_1|^{A_1} cdots |x_n|^{A_n}$ in $mathbb{R}^n$ , where $A_i geq 0$, we can establish the weighted Sobolev, Morrey, and Trudinger inequalities with such weights [Cabre and Ros-Oton, 2013].

並列關鍵字

Isoperimetric inequality ； ABP method ； Neumann problem ； Sobolev inequality ； monomial weight.

參考文獻

Xavier Cabre, Xavier Ros-Oton, and Joaquim Serra. Sharp isoperimetric inequalities via the abp method. arXiv preprint arXiv:1304.1724, 2013.

Xavier Cabre and Xavier Ros-Oton. Sobolev and isoperimetric inequalities with monomial weights. Journal of Differential Equations, 255(11):4312–4336, 2013.

R Gardner. The brunn-minkowski inequality. Bulletin of the American Mathematical Society, 39(3):355–405, 2002.

Georg Wulff. Zur frage der geschwindigkeit des wachstums und der auflosung der kristallflachen. Z. kristallogr, 34(5/6):449–530, 1901.

Jean E Taylor. Existence and structure of solutions to a class of nonelliptic variational problems. In Symposia Mathematica, volume 14, pages 499–508, 1974.