Multiphase quadrature domains (existence and uniqueness)

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 proving 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. However, we also show that this approach alone is not enough for two reasons. First it seems difficult to obtain 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, when they exist, of the energy functional over a subset of the segregated states determined by a natural constraint associated with the prescribed 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.