The Bergman analytic content of planar domains

TL;DR

This paper computes the Bergman analytic content for certain planar domains, linking it to torsional rigidity in elasticity theory, and explores differences between simply-connected and multiply-connected domains.

Contribution

It provides explicit calculations of Bergman analytic content for quadrature domains and establishes its relation to torsional rigidity, highlighting differences based on domain connectivity.

Findings

Bergman analytic content computed for quadrature domains.

Square of analytic content is equivalent to torsional rigidity in simply-connected domains.

In multiply-connected domains, these constants are not equivalent.

Abstract

Given a planar domain $\Omega$, the Bergman analytic content measures the $L^{2}(\Omega)$-distance between $\bar{z}$ and the Bergman space $A^{2}(\Omega)$. We compute the Bergman analytic content of simply-connected quadrature domains with quadrature formula supported at one point, and we also determine the function $f \in A^2(\Omega)$ that best approximates $\bar{z}$. We show that, for simply-connected domains, the square of Bergman analytic content is equivalent to torsional rigidity from classical elasticity theory, while for multiply-connected domains these two domain constants are not equivalent in general.