Regularity of a free boundary with application to the Pompeiu problem

In the unit ball B(0, 1), let u and Omega (a domain in R-N) salve the follo wing overdetermined problem: Delta u = chi(Omega) in B(0, 1), 0 is an element of partial derivative Omeg a, u = \del u\ = 0 in B(0, 1) \ Omega, where chi(Omega) denotes the characteristic function, and the equation is s atisfied in the sense of distributions. If the complement of Omega does not develop cusp singularities at the origi n then we prove partial derivative Omega is analytic in some small neighbor hood of the origin. The result can be modified to yield for more general di vergence form operators. As an application of this, then, we obtain the reg ularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.