[Paper Review] Analytic Capacity : computation and related problems

Malik Younsi|arXiv (Cornell University)|May 8, 2017

Analytic and geometric function theory12 references3 citations

TL;DR

This paper presents a rigorous numerical method for computing analytic capacity using quadratic minimization, yielding tight upper and lower bounds. It establishes connections to subadditivity, Ahlfors functions, and Cauchy capacity, and resolves open problems regarding capacity convergence and measure representation.

ABSTRACT

We present a brief introduction to analytic capacity, with an emphasis on its numerical computation. We also discuss several related open problems.

Motivation & Objective

• To develop a rigorous, efficient numerical method for computing analytic capacity of compact sets.
• To investigate the subadditivity of analytic capacity and reduce it to a special case involving finite unions of equal-radius disjoint disks.
• To determine whether the Ahlfors function for a compact set can be represented as a Cauchy transform of a complex Borel measure.
• To explore the relationship between analytic capacity and Cauchy capacity, particularly in cases of infinite length or non-rectifiable curves.
• To address open problems on inner-regularity and attainment of extremal measures for related capacities.

Proposed method

• The method uses quadratic minimization to compute rigorous upper and lower bounds for analytic capacity, converging to the true value.
• It relies on a discretization of the extremal problem via bounded analytic functions with controlled $ L^lat $-norms.
• The approach leverages weak* compactness and weak* convergence to ensure convergence of approximations.
• The algorithm is applied numerically to test subadditivity and validate conjectures on capacity behavior.
• Conformal invariance and convergence theorems (e.g., Pommerenke’s) are used to analyze limits of capacity under set approximation.
• The method is extended in principle to compute related capacities $ ilde{ u}_+ $ and $ ilde{ u}_c $, though their attainment properties remain open.

Experimental results

Research questions

• RQ1 Does the subadditivity inequality $ ilde{ u}(E igcup F) eq ilde{ u}(E) + ilde{ u}(F) $ hold for all compact sets $ E, F $, and can it be reduced to a finite union of equal-radius disjoint disks?
• RQ2 Can the Ahlfors function for a compact set $ E $ be represented as the Cauchy transform of a complex Borel measure supported on $ E $?
• RQ3 Is analytic capacity inner-regular, i.e., does $ ilde{ u}(E_n) \uparrow \tilde{ u}(E) $ when $ E_n \uparrow E $?
• RQ4 Does the positive capacity $ \tilde{\nu}_+ $ always attain its supremum, and is the maximizer unique?
• RQ5 Does $ \tilde{\nu}(F_k) \to \tilde{\nu}(F) $ when $ F_k $ is a union of $ n \geq 3 $ disks converging to a union intersecting at a point?

Key findings

• The numerical method from [28] provides rigorous upper and lower bounds for analytic capacity that converge to the true value.
• Subadditivity of analytic capacity reduces to the case of finite unions of disjoint closed disks of equal radius.
• There exists a connected compact set $ E $ with connected complement such that $ \tilde{\nu}(E) = \tilde{\nu}_c(E) $, but the Ahlfors function is not the Cauchy transform of any complex measure on $ E $.
• The set $ E = \{x + i x \sin(1/x) : x \in (0,1/\pi]\} \cup [-i,i] $ has infinite length and satisfies $ \tilde{\nu}(E) = \tilde{\nu}_c(E) $, demonstrating equality despite non-rectifiability.
• For sequences $ E_k $ of finite-length sets increasing to $ E $, $ \tilde{\nu}(E_k) \to \tilde{\nu}(E) $, supporting convergence results for analytic capacity.
• The capacity $ \tilde{\nu}_+ $ is not known to be attained by a unique measure, and this remains an open problem.