Multiphase quadrature domains exist and are unique under sufficient conditions via constrained minimization of an energy functional over segregated states, with an example showing that energy minimization and partial balayage are not equivalent in the two-phase case.
The results in the appendix are well-known, and the proofs can found at
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Multiphase quadrature domains (existence and uniqueness)
Multiphase quadrature domains exist and are unique under sufficient conditions via constrained minimization of an energy functional over segregated states, with an example showing that energy minimization and partial balayage are not equivalent in the two-phase case.