  Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0
\textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a
realistic class of solutions, containing for example {\em all} limits of the
singular perturbation problem $$\Delta u_\epsilon - \partial_t u_\epsilon =
\beta_\epsilon(u_\epsilon) \textrm{as} \epsilon\to 0,$$ we prove that one-sided
flatness of the free boundary implies regularity.
  In particular, we show that the topological free boundary $\partial\{u>0\}$
can be decomposed into an {\em open} regular set (relative to
$\partial\{u>0\}$) which is locally a surface with H\"older-continuous space
normal, and a closed singular set.
  Our result extends the main theorem in the paper by H.W. Alt-L.A. Caffarelli
(1981) to more general solutions as well as the time-dependent case. Our proof
uses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replace
the core of that paper, which relies on non-positive mean curvature at singular
points, by an argument based on scaling discrepancies, which promises to be
applicable to more general free boundary or free discontinuity problems.
