some properties of two-phase quadrature domains

in this paper, we investigate general properties of the two-phase quadrature domain, which was recently introduced by emamizadeh, prajapat and shahgholian. the concept, which is a generalization of the well-known one-phase domain, introduces substantial difficulties with interesting features even richer than those of the one-phase counterpart. for given positive constants λ± and two bounded and compactly supported measures μ±, we investigate the uniqueness of the solution of the following free boundary problem:

{ δu = (λ⁺χ_ω⁺ − μ⁺) − (λ⁻χ_ω− − μ⁻), in rⁿ (n ≥ 2),

u = 0, in rⁿ \ ω,(1)

 where ω = ω⁺ ∪ ω⁻. it is further required that the supports of μ± should be inside ω^±; this in general may fail and give rise to non-existence of solutions. along the paths to various properties that we state and prove here, we also present several conjectures and open problems that we believe should be true.