论文标题
离散重排和图形上的polya-szego不平等
Discrete Rearrangements and the Polya-Szego Inequality on Graphs
论文作者
论文摘要
对于任何$ f：\ mathbb {r}^n \ rightarrow \ mathbb {r} _ {\ geq 0} $对称减小重排$ f^*$满足Polya-Szegő不平等$ \ | \ nabla f^*\ | _ {l^p} \ leq \ | \ nabla f \ | _ {l^p} $。本文的目的是在满足适当条件的图表的离散设置中建立类似的结果。我们证明，如果图上的边缘 - 术问题具有一系列嵌套的最小化器，那么该序列会产生重排，使Polya-Szegő不平等达到$ l^1 $。例如，这表明网格图上的特定重排$ \ mathbb {z}^2 $，以类似螺旋的方式绕起来，满足$ \ | \ nabla f^*\ | _ {l^1} \ leq \ | \ nabla f \ | _ {l^1} $。 $ l^{\ infty} - $ case在顶点 - 等法中的最佳订购条件暗示。我们使用这些想法来证明无限$ d $常规树上的规范重排满足了Polya-szegő不平等，所有$ 1 \ leq p \ leq \ leq \ infty $。
For any $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ the symmetric decreasing rearrangement $f^*$ satisfies the Polya-Szegő inequality $\| \nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$. The goal of this paper is to establish analogous results in the discrete setting for graphs satisfying suitable conditions. We prove that if the edge-isoperimetric problem on a graph has a sequence of nested minimizers, then this sequence gives rise to a rearrangement satisfying the Polya-Szegő inequality in $L^1$. This shows, for example, that a specific rearrangement on the grid graph $\mathbb{Z}^2$, going around the origin in a spiral-like manner, satisfies $\| \nabla f^*\|_{L^1} \leq \| \nabla f\|_{L^1}$. The $L^{\infty}-$case is implied by an optimal ordering condition in vertex-isoperimetry. We use these ideas to prove that the canonical rearrangement on the infinite $d-$regular tree satisfies the Polya-Szegő inequality for all $1 \leq p \leq \infty$.
