Null quadrature domains and a free boundary problem for the Laplacian

TL;DR

This paper studies null quadrature domains, proving their complements are convex with real analytic boundaries, and connects these properties to classical free boundary problems for the Laplacian, revealing geometric and regularity characteristics.

Contribution

It establishes the convexity and real analyticity of null quadrature domain complements and links their properties to classical free boundary problem regularity results.

Findings

Complement of null quadrature domains is convex with real analytic boundary.

Null quadrature domains with non-zero density at infinity are half-spaces.

Quadratic growth estimate for the Schwarz potential is proved.

Abstract

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.