Computational Methods and Function Theory 5 (2005), No. 1, 19--48
Copyright Heldermann Verlag 2005
Dmitry Khavinson
dmitry@uark.edu , Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, U.S.A.
Alexander Yu. Solynin
solynin@math.ttu.edu , Department of Mathematics and Statistics, Texas Tech University, P.O. Box 41042 Lubbock, Texas 79409-1042, U.S.A.
Dimiter Vassilev
vassilev@math.uqam.ca , CRM/ISM -- UQAM, C.P. 8888, succursale Centre-Ville, Montréal, PQ, H3C 3P8, Canada.
We discuss an overdetermined problem in planar multiply connected domains $\Omega$. This problem is solvable in $\Omega$ if and only if $\Omega$ is a quadrature domain carrying a solid-contour quadrature identity for analytic functions. At the same time the existence of such quadrature identity is equivalent to the solvability of a special boundary value problem for analytic functions. We give a complete solution of the problem in some special cases and discuss some applications concerning the shape of electrified droplets and small air bubbles in a fluid flow.
Keywords: Overdetermined boundary value problem, quadrature domain, analytic function, quadratic differential.
MSC 2000: Primary 30E25; Secondary 30F30, 76D27, 35A30, 35R35.
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