Note about the existence and essential uniqueness of quadrature domains

TL;DR

This paper discusses the existence and essential uniqueness of quadrature domains, providing simplified proofs for these properties and demonstrating their existence for a broad class of weight functions.

Contribution

It offers an accessible proof of the uniqueness and existence of quadrature domains, expanding understanding of their fundamental properties.

Findings

Quadrature domains are essentially unique if they exist.

They exist for a large class of weight functions.

Simplified proofs of key properties are provided.

Abstract

This note is intended to explain the proof of two facts about quadrature domains: first, they are essentially unique if they exist; and second, they do exist for a large class of weight functions. The proofs roughly follow Sakai's "Solutions to the obstacle problem as Green potentials," but are presented at an easier level.