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arXiv · 1512.06577

The annular decay property and capacity estimates for thin annuli

Anders Björn·Jana Björn·Juha Lehrbäck
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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 Poincaré inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincaré 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.

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Anders Björn, Jana Björn, Juha Lehrbäck. 2015-12-21. The annular decay property and capacity estimates for thin annuli. https://doi.org/10.1007/s13348-016-0178-y

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