Continuity of logarithmic capacity

TL;DR

This paper proves that logarithmic capacity remains continuous under Hausdorff convergence for uniformly perfect planar sets, with specific conditions on the convergence rate, highlighting cases where continuity may fail.

Contribution

It establishes the conditions under which logarithmic capacity is continuous during Hausdorff convergence of uniformly perfect sets, clarifying when this property holds or fails.

Findings

Continuity of logarithmic capacity is guaranteed under certain convergence rates.

Continuity may fail if the convergence rate is too slow.

Provides conditions relating Hausdorff convergence and uniformly perfect sets.

Abstract

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise.