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 \begin{document}$\int_{\partial Ω^+} g h (x) \ dσ_x - \int_{\partial Ω^-} g h (x) \ dσ_x= \int h dμ \ ,$ \end{document}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.
Highlights
Where dσx is the surface measure, μ = μ+ − μ− is given measure with support in Ω, g is a given smooth positive function, and the integral holds for all functions h, which are harmonic on 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
Summary
The current paper concerns the so-called quadrature identities for surface integrals, for the harmonic class of functions, and for given measures. The free boundary communities, specially those working with regularity theory, would find an interesting extension of the concept of two-phase Bernoulli problem, with the zero set having non-void interior. This obviously makes the problem a three phase problem with the third phase being free of fluid. Quadrature domains can be obtained as supports of local minimizers for the one phase functional (1.9). If we let h = 0 ν is the surface measure, and solution to this problem corresponds to (one phase) quadrature surfaces.
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