Uniformly Perfect Sets, Hausdorff Dimension, and Conformal Capacity




Rainio Oona, Sugawa Toshiyuki, Vuorinen, Matti

PublisherSpringer Nature

2024

Journal of Geometric Analysis

JOURNAL OF GEOMETRIC ANALYSIS

ARTN 154

34

6

1050-6926

1559-002X

DOIhttps://doi.org/10.1007/s12220-024-01599-5(external)

https://link.springer.com/article/10.1007/s12220-024-01599-5(external)

https://research.utu.fi/converis/portal/detail/Publication/393346031(external)



Using the definition of uniformly perfect sets in terms of convergent sequences, we apply lower bounds for the Hausdorff content of a uniformly perfect subset E of Rn to prove new explicit lower bounds for the Hausdorff dimension of E. These results also yield lower bounds for capacity test functions, which we introduce, and enable us to characterize domains of Rn with uniformly perfect boundaries. Moreover, we show that an alternative method to define capacity test functions can be based on the Whitney decomposition of the domain considered.


Last updated on 2024-26-11 at 14:12