The annular decay property and capacity estimates for thin annuli
2016; Springer Science+Business Media; Volume: 68; Issue: 2 Linguagem: Inglês
10.1007/s13348-016-0178-y
ISSN2038-4815
AutoresAnders Björn, Jana Björn, Juha Lehrbäck,
Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoWe 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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