The annular decay property and capacity estimates for thin annuli

• Autores: Anders Björn, Jana Björn, Juha Lehrbäck
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 68, Fasc. 2, 2017, págs. 229-241
• Idioma: inglés
• DOI: 10.1007/s13348-016-0178-y
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• Resumen

  • 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.