Non-algebraic quadrature domains

Abstract

It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial. We show, by explicitly constructing some four-dimensional examples, that this is not always the case. This confirms, in dimension 4, a conjecture of the second author. Our method is based on the Schwarz potential and involves elliptic integrals of the third kind.

23 pages, 4 figures

References in corpus (2)

• The Universality Class of Diffusion Limited Aggregation and Viscous Fingering
• Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

Cited by in corpus (2)

• Torsional Rigidity and Bergman Analytic Content of Simply Connected Regions
• On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth