Point Source Equilibrium Problems with Connections to Weighted Quadrature Domains

TL;DR

This paper investigates the relationship between equilibrium measure supports and quadrature identities, focusing on point sources in specific external fields and describing related weighted quadrature domains.

Contribution

It establishes new connections between equilibrium measures and weighted quadrature domains, particularly for external fields of the form |z|^{2p} with p in natural numbers.

Findings

Characterization of quadrature domains with respect to weighted area measure

Description of complex boundary measures for these domains

Analysis of equilibrium measures in the presence of point sources

Abstract

We explore the connection between supports of equilibrium measures and quadrature identities, especially in the case of point sources added to the external field $Q(z)=|z|^{2p}$ with $p \in \mathbb{N}$. Along the way, we describe some quadrature domains with respect to weighted area measure $|z|^{2p}dA_z$ and complex boundary measure $|z|^{-2p}dz$.