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Two- and Multi-phase Quadrature Surfaces

In this paper we shall initiate the study of the two- and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation $$ \int_{\partial Ω^+} g h (x) \ dσ_x - \int_{\partial Ω^-} g h (x) \ dσ_x= \int h dμ\ , $$ where $dσ_x$ is the surface measure, $μ= μ^+ - μ^-$ is given measure with support in (a priori unknown domain) $Ω$, $g$ is a given smooth positive function, and the integral holds for all functions $h$, which are harmonic on $\overline Ω$. Our approach is based on minimization of the corresponding two- and multi-phase functional and the use of its one-phase version as a barrier. We prove several results concerning existence, qualitative behavior, and regularity theory for solutions. A central result in our study states that three or more junction points do not appear.