article

Title: Null quadrature domains and a free boundary problem for the Laplacian

Authors: Lavi Karp and Avmir Margulis

Issue: Volume 61 (2012), Issue 2, 859-882

Abstract:

$Null quadrature domains are unbounded domains in $\mathbb{R}^n$ ($n \geq 2$) with external gravitational force zero in some generalized sense. In this paper we prove that the complement of a 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 [L.A. Caffarelli, L. Karp, and H. Shahgholian, \textit{Regularity of a free boundary with application to the Pompeiu problem}, Ann. of Math. (2) \textbf{151} (2000), no. 1, 269--292] on the regularity of the solution to the classical global free boundary problem for the Laplacian. We also show that any null quadrature domain with non-zero upper Lebesgue density at infinity is a half-space.$