  In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\R$) solve the
following overdetermined problem: $$\Delta u =\chi_\Omega\quad \hbox{in}
B(0,1), \qquad 0 \in \partial \Omega, \qquad u=|\nabla u |=0 \quad \hbox{in}
B(0,1)\setminus \Omega,$$ where $\chi_\Omega$ denotes the characteristic
function, and the equation is satisfied in the sense of distributions. If the
complement of $\Omega$ does not develop cusp singularities at the origin then
we prove $\partial \Omega$ is analytic in some small neighborhood of the
origin. The result can be modified to yield for more general divergence form
operators. As an application of this, then, we obtain the regularity of the
boundary of a domain without the Pompeiu property, provided its complement has
no cusp singularities.
