On the free boundary regularity theorem of Alt and Caffarelli

Abstract

In this note we discuss a slight generalization of the following result by Alt and Caffarelli: if the logarithm of the Poisson kernel of a Reifenberg flat chord arc domain is Hölder continuous, then the domain can be locally represented as the area above the graph of a function whose gradient is Hölder continuous. In this note we show that if the Poisson kernel of an unbounded Reifenberg flat chord arc domain is 1 a.e. on the boundary then the domain is (modulo rotation and translation) the upper half plane. This result plays a key role in the study of regularity of the free boundary below the continuous threshold.