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Two-and multi-phase quadrature surfaces

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A. Arakelyan was supported by State Committee of Science MES RA, in frame of the research project No. 16YR-1A017. H. Shahgholian is partially supported by the Swedish Research Council

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  • 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) $Ω=Ω^+\cupΩ^-$ , $g$ 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.

    Mathematics Subject Classification: Primary: 35R35, 31A05, 31B05, 31B20.


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