Two- and Multi-phase Quadrature Surfaces

TL;DR

This paper introduces the study of two- and multi-phase quadrature surfaces as free boundary problems of Bernoulli type, establishing existence, regularity, and the non-occurrence of junction points for three or more phases.

Contribution

It develops a potential theoretic framework for multi-phase quadrature surfaces, proving key properties and the non-existence of complex junction points.

Findings

Established existence and regularity of solutions.

Proved that three or more junction points do not occur.

Developed a minimization approach using one-phase solutions as barriers.

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.