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±λ± and two bounded and compactly supported measures μ±μ±, we investigate the uniqueness of the solution of the following free boundary problem: equation(1){Δu=(λ+χΩ+−μ+)−(λ−χΩ−−μ−),in RN(N≥2),u=0,in RN∖Ω, where Ω=Ω+∪Ω−Ω=Ω+∪Ω−. It is further required that the supports of μ±μ± should be inside Ω±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.