Null quadrature domains and a free boundary problem for the Laplacian

Null quadrature domains are unbounded domains in $\R^n$ ($n \geq 2$) with external gravitational force zero in some generalized sense. In this paper we prove that the complement of null quadrature domain is a convex set with real analytic boundary. We establish the quadratic growth estimate for the Schwarz potential of a null quadrature domain which reduces our main result to Theorem II of the paper of Caffarelli, Karp and Shahgholian (Ann. Math. 151(2000), 269-292), on the regularity of solution to the classical global free boundary problem for Laplacian. We also show that any null quadrature domain with non-zero upper Lebesgue density at infinity is half-space.