Overdetermined Boundary Value Problems, Quadrature Domains and Applications

We discuss an overdetermined problem in planar multiply connected domains $\Om$. This problem is solvable in $\Om$ if and only if $\Om$ 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.