The annular decay property and capacity estimates for thin annuli

TL;DR

This paper establishes sharp bounds for the nonlinear variational capacity of thin annuli in weighted Euclidean spaces and metric spaces, generalizing classical estimates and characterizing the annular decay property.

Contribution

It provides new two-sided estimates for variational capacity under annular decay and Poincaré conditions, extending known results to weighted and metric settings.

Findings

Bounds are sharp, supported by counterexamples.

Capacity estimates are comparable under annular decay and Poincaré conditions.

Characterization of the 1-annular decay property.

Abstract

We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also characterize the $1$-annular decay property.