On null quadrature domains
- 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 done by Friedman and Sakai ; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellip- soids. The first result with unbounded complements appeared in , 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 succeed to make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a strip and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a strip. In addition, we present a Phragm¶en-Lindelöf type theorem which seems to be forgotten in the literature.