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arXiv · 2209.06765

Discrete Rearrangements and the Polya-Szego Inequality on Graphs

Stefan Steinerberger
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Abstract

For any $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ the symmetric decreasing rearrangement $f^*$ satisfies the Polya-Szeg\H{o} 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-Szeg\H{o} 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-Szeg\H{o} inequality for all $1 \leq p \leq \infty$.

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Stefan Steinerberger. 2022-09-14. Discrete Rearrangements and the Polya-Szego Inequality on Graphs. https://arxiv.org/abs/2209.06765

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