An Exponential Transform and Regularity of Free Boundaries in Two Dimensions

Abstract

We investigate the basic properties of the exponential transform E(z, w) = exp 7r 0 x dA(~) -) ) (z, w E C) of a domain Q c C and compute it in some simple cases. The main result states that if the Cauchy trans- form of the characteristic function of Q has an analytic continuation from C B S2 across aS2 then the same is true for E(z, w), in both variables. If F (z, w) de- notes this analytic-antianalytic continuation it follows that 8 Q is contained in a real analytic set, namely the zero set of F (z, z). This gives a new approach to the regularity theory for free boundaries in two dimensions.