Two- and Multi-phase Quadrature Surfaces

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 $${\int }_{\partial {\Omega }^{+}}gh(x)d{\sigma }_x-{\int }_{\partial {\Omega }^{-}}gh(x)d{\sigma }_x=\int hd\mu ,$$ 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.

arXiv comment28 pages, Keywords: two-phase quadrature surface, free boundary, Bernoulli boundary condition