# American Institute of Mathematical Sciences

November  2017, 16(6): 2023-2045. doi: 10.3934/cpaa.2017099

 1 Institute of Mathematics National Academy of Sciences of Armenia, 0019 Yerevan, Armenia 2 Department of Mathematics Royal Institute of Technology, 100 44 Stockholm, Sweden 3 Department of Mathematics, University of Mumbai Vidyanagari, Santacruz (east), 400 097 Mumbai, India

* Corresponding author

Received  September 2016 Revised  December 2016 Published  July 2017

Fund Project: 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.

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.
Citation: Avetik Arakelyan, Henrik Shahgholian, Jyotshana V. Prajapat. Two-and multi-phase quadrature surfaces. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2023-2045. doi: 10.3934/cpaa.2017099
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