Two-and multi-phase quadrature surfaces

Avetik Arakelyan; Henrik Shahgholian; Jyotshana V. Prajapat

doi:10.3934/cpaa.2017099

Article Contents

2017, Volume 16, Issue 6: 2023-2045. Doi: 10.3934/cpaa.2017099

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Two-and multi-phase quadrature surfaces

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
* Corresponding author

Received: September 2016

Revised: December 2016

Published: September 2017

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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Abstract

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.

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

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