Computation of Analytic Capacity and applications to the Subadditivity Problem

TL;DR

This paper introduces a least-squares computational method for accurately estimating the analytic capacity of certain plane sets, providing bounds that converge to the true value, and explores implications for the subadditivity conjecture.

Contribution

A novel least-squares approach for computing analytic capacity with rigorous bounds and a new conjecture related to subadditivity, proved in a special case.

Findings

Method yields converging bounds for analytic capacity

Conjecture on subadditivity formulated and partially proved

Illustrative examples demonstrate effectiveness

Abstract

We develop a least-squares method for computing the analytic capacity of compact plane sets with piecewise-analytic boundary. The method furnishes rigorous upper and lower bounds which converge to the true value of the capacity. Several illustrative examples are presented. We are led to formulate a conjecture which, if true, would imply that analytic capacity is subadditive. The conjecture is proved in a special case.