Sharp capacity estimates in s-John domains

TL;DR

This paper establishes new lower bounds for p-capacity in s-John domains using Hausdorff content, providing sharp estimates and examples to demonstrate their optimality.

Contribution

It introduces general lower bounds for p-capacity in s-John domains based on Hausdorff content, advancing understanding of capacity estimates in these domains.

Findings

Derived sharp lower bounds for p-capacity in s-John domains.

Connected capacity estimates with Hausdorff q-content of sets.

Provided examples demonstrating the sharpness of the bounds.

Abstract

It is well-known that several problems related to analysis on $s$-John domains can be unified by certain capacity lower estimates. In this paper, we obtain general lower bounds of $p$-capacity of a compact set $E$ and the central Whitney cube $Q_0$ in terms of the Hausdorff $q$-content of $E$ in an $s$-John domain $\Omega$. Moreover, we construct several examples to show the essential sharpness of our estimates.