Algebraicity in the Dirichlet problem in the plane with rational data

Abstract

We consider the Dirichlet problem for the Laplace operator with rational data on the boundary of a planar domain. Our main results include a characterization of the disk as the only domain for which all solutions are rational, and a characterization of the simply connected quadrature domains as the only ones for which all solutions are algebraic of a certain type. This note is an exposition, and full details will appear in a forthcoming paper.

Keywords:

• Dirichlet problem
• Quadrature domain
• Laplac operator

Keywords:

• 1991 Mathematics Subject Classifications: 30E34
• 35J25

Acknowledgements

The authors Bell, Ebenfelt, and Khavinson were partially supported by the NSF grants DMS-0305958, DMS-0401215, and DMS-0139008 respectively.

Notes

∥Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.