Two- and Multi-phase Quadrature Surfaces

Avetik Arakelyan; Jyotshana V. Prajapat; Henrik Shahgholian

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arXiv · 1610.02637

Two- and Multi-phase Quadrature Surfaces

Avetik Arakelyan·Jyotshana V. Prajapat·Henrik Shahgholian

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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 $$ \int_{\partial \Omega^+} g h (x) \ d\sigma_x - \int_{\partial \Omega^-} g h (x) \ d\sigma_x= \int h d\mu \ , $$ where $d\sigma_x$ is the surface measure, $\mu= \mu^+ - \mu^-$ is given measure with support in (a priori unknown domain) $\Omega$, $g$ is a given smooth positive function, and the integral holds for all functions $h$, which are harmonic on $\overline \Omega$. 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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Avetik Arakelyan, Jyotshana V. Prajapat, Henrik Shahgholian. 2016-10-09. Two- and Multi-phase Quadrature Surfaces. https://arxiv.org/abs/1610.02637

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