In the upper half of the unit ball $B\sp + =\{ |x|<1,x\sb
1>0\}$, let $u$ and $\Omega$ (a domain in $\mathbf {R}\sp n\sb +
=\{x\in \mathbf {R}\sp n : x\sb 1>0\}$) solve the following
overdetermined problem:
\Delta u =\chi_\Omega\quad \text{in}\ B^+,
\qquad u=|\nabla u |=0 \quad \text{in}\ B^+\setminus \Omega,
\qquad u=0 \quad \text{on}\ \Pi\cap B,
¶ where $B$ is the unit ball with center at the origin, $\chi\sb
\Omega$ denotes the characteristic function of $\Delta,\Pi=\{ x\sb
1=0\} ,n\geq 2$, and the equation is satisfied in the sense of
distributions. We show (among other things) that if the origin is a
contact point of the free boundary, that is, if $u(0)=|\nabla
u(0)|=0$, then $\partial\Delta\cap B\sb {r\sb 0}$ is the graph of a
$C\sp 1$-function over $\Pi\cap B\sb (r\sb 0)$. The $C\sp 1$-norm
depends on the dimension and $\sup\sb {B\sp +}|u|$. The result is
extended to general subdomains of the unit ball with $C\sp
3$-boundary.