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arXiv · 2311.11887

An Almgren monotonicity formula for discrete harmonic functions

Mariana Smit Vega Garcia·Stefan Steinerberger
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Abstract

The celebrated Almgren monotonicity formula for harmonic functions $u:\mathbb{R}^n \rightarrow \mathbb{R}$ says that its $L^2-$energy concentrated on a sphere of radius $r$, when measured in a suitable sense, is non-decreasing: if $u$ oscillates at a certain scale, it has even larger oscillations at a larger scale. We prove a discrete analogue of the Almgren monotonicity formula for harmonic functions on infinite combinatorial graphs $G=(V,E)$. Some applications are discussed.

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Mariana Smit Vega Garcia, Stefan Steinerberger. 2023-11-20. An Almgren monotonicity formula for discrete harmonic functions. https://arxiv.org/abs/2311.11887

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