On the free boundary regularity theorem of Alt and Caffarelli. (Q1433929)

This paper generalizes a result of 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 represented as the area above the graph of a function whose gradient is Hölder continuous. The authors prove that in unbounded domains of the same type, if the Poisson kernel is 1 on the boundary, then the domain is the half space. The Reifenberg regularity condition, which occurs in several different contexts, roughly speaking requires that the boundary can be uniformly locally approximated by codimension 1 hyperplanes.