Multiphase quadrature domains (existence and uniqueness)

TL;DR

This paper establishes existence and uniqueness results for multiphase quadrature domains for subharmonic functions, using a variational approach and analyzing minimizers under natural constraints, highlighting differences from one-phase cases.

Contribution

It introduces a novel variational framework for studying multiphase quadrature domains, proving uniqueness and providing conditions for existence, and clarifies differences from one-phase scenarios.

Findings

Proved existence and uniqueness of multiphase quadrature domains.

Identified that energy minimizers may not always correspond to quadrature domains.

Showed the breakdown of equivalence between energy minimization and partial balayage in the multiphase case.

Abstract

The primary goal of this paper is to give a precise definition and prove existence and uniqueness of multiphase quadrature domains for subharmonic functions, ensuring that the prescribed measures are supported in the interior of the resulting domains. The approach to prove existence is based on a variational framework, where we minimize an energy functional over so called segregated states. In this respect we refine earlier results in this direction. But we also show that this approach alone is not enough for two reasons. First of all it seems hard to get existence results which ensure that the interior support condition is satisfied. And second it may happen, as we show by an example, that a multiphase quadrature domain exists but is not a minimizer of the energy functional. The main novelty of this work is the study of minimizers, if they exist, of the energy functional over a subset of the segregated states given by a natural constraint with respect to the given measures. From this approach we are able to prove uniqueness, and also give sufficient conditions for existence. We also give an example showing that, unlike the energy minimization and partial balayage approaches which are equivalent in the one-phase case, this equivalence breaks down already in the two-phase setting.