Overdetermined Boundary Value Problems, Quadrature Domains and Applications

Abstract. We discuss an overdetermined problem in planar multiply con-nected domains Ω. This problem is solvable in Ω if and only if Ω 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 ap-plications concerning the shape of electrified droplets and small air bubbles in a fluid flow