Computational Methods and Function Theory 8 (2008), No. 1, 57--72
Copyright Heldermann Verlag 2008
Lavi Karp
lavikarp@gmail.com , ORT Braude College, Department of Mathematics, P.O. Box 78, 21982 Karmiel, Israel.
The characterization of null quadrature domains in Rn, n≥3, has been an open problem throughout the past two and a half decades. A substantial contribution was made by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellipsoids. The first result with unbounded complements appeared in [16], there it is assumed the complement is contained in an infinitely cylinder. The aim of this paper is to show the relation between null quadrature domains and Newton's theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a slab and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a slab. In addition, we present a Phragmén-Lindelöf type theorem which we could not find in the literature.
Keywords: Null quadrature domains, homoeoidal ellipsoid, gravitational attraction, Newtonian potential, Newtonian capacity.
MSC 2000: Primary 31C15, 35J25; Secondary 35B50, 31B15.
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