Multi-phase Quadrature domains and a related minimization problem

TL;DR

This paper introduces the multi-phase Quadrature Domains, extending the classical mean value property for harmonic functions to multiple phases, and establishes existence and properties of these solutions, including multi-junction points.

Contribution

It generalizes the theory of Quadrature Domains from one- and two-phase cases to multiple phases, providing foundational results and discussing complex junction points.

Findings

Existence of multi-phase Quadrature Domains proven.

Properties of solutions including multi-junction points analyzed.

Extension of classical QD theory to multi-phase scenarios.

Abstract

In this paper we introduce the multi-phase version of the so-called Quadrature Domains (QD), which refers to a generalized type of mean value property for harmonic functions. The well-established and developed theory of one-phase QD was recently generalized to a two-phase version, by one of the current authors (in collaboration). Here we introduce the concept of the multi-phase version of the problem, and prove existence as well as several properties of such solutions. In particular, we discuss possibilities of multi-junction points.