Quadrature domains packing

TL;DR

This paper introduces a criterion based on positivity conditions of an analytic kernel to certify non-overlapping of compact sets in the complex plane, with applications to quadrature domains and an effective matrix algorithm.

Contribution

It proposes a novel positivity-based certificate for mutual non-overlapping of compact sets, especially quadrature domains, using an analytic kernel and matrix analysis.

Findings

Positivity conditions effectively certify non-overlapping.

Rational kernels enable matrix algorithms for quadrature domains.

Detailed analysis of two disks case from matrix and potential theory perspectives.

Abstract

Given a finite family of compact subsets of the complex plane we propose a certificate of mutual non-overlapping with respect to area measure. The criterion is stated as a couple of positivity conditions imposed on a four argument analytic/anti-analytic kernel defined in a neighborhood of infinity. In case the compact sets are closures of quadrature domains the respective kernel is rational, enabling an effective matrix analysis algorithm for the non-overlapping decision. The simplest situation of two disks is presented in detail from a matrix model perspective as well as from a Riemann surface potential theoretic interpretation.