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The annular decay property and capacity estimates for thin annuli

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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 \(x_0\). This generalizes the known estimate for the usual variational capacity in unweighted \(\mathbf {R}^n\). We also characterize the 1-annular decay property and provide examples which illustrate the sharpness of our results.

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Acknowledgments

A. B. and J. B. were supported by the Swedish Research Council. J. L. was supported by the Academy of Finland (Grant No. 252108) and the Väisälä Foundation of the Finnish Academy of Science and Letters. Part of this research was done during several visits of J. L. to Linköping University in 2012–15, and one visit of A. B. to the University of Jyväskylä in 2015.

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Anders Björn & Jana Björn

  2. Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35, 40014, Jyväskylä, Finland

    Juha Lehrbäck

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  1. Anders Björn
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  2. Jana Björn
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  3. Juha Lehrbäck
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Correspondence to Juha Lehrbäck.

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Björn, A., Björn, J. & Lehrbäck, J. The annular decay property and capacity estimates for thin annuli. Collect. Math. 68, 229–241 (2017). https://doi.org/10.1007/s13348-016-0178-y

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