Quadrature domains

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

Quadrature domains are a special kind of region in the plane with a simple averaging property for harmonic functions.

Take a region D in the plane and a finite set of points {z1, ..., zk} inside D. If every harmonic function u that is integrable over D satisfies that its total integral over D equals a fixed weighted sum of the values of u at the points z1, ..., zk (the weights are fixed numbers that do not depend on u), then D with these points is a quadrature domain.

The simplest example is a circular disk: here k = 1, the single point is the center, and the weight equals the area of D. This reflects the mean value property for harmonic functions in disks.

Quadrature domains exist for any number k of special points. A similar idea works in higher dimensions.

There is also an electrostatic interpretation: if you place a uniform charge on D, the resulting electric field outside D is the same as if you had k point charges placed at z1, ..., zk.

These domains and related ideas appear in various areas, such as inverse problems in Newtonian gravitation, flows of viscous fluids in Hele-Shaw cells, and mathematical isoperimetric questions. Interest in them has been growing, including an international conference on the topic at the University of California, Santa Barbara, in 2003; the latest work from that meeting was published in its conference volume.