Weighted Isoperimetric Inequalities via the ABP method and its Applications
Date Issued
2014
Date
2014
Author(s)
Lin, Kuan-Yu
Abstract
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].
Subjects
等周不等式
ABP方法
Neumann問題
Sobolev不等式
單項式權重.
Type
thesis
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