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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 arealistic class of solutions, containing for example {\em all} limits of thesingular perturbation problem $$\Delta u_\epsilon - \partial_t u_\epsilon =\beta_\epsilon(u_\epsilon) \textrm{as} \epsilon\to 0,$$ we prove that one-sidedflatness 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 spacenormal, 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 proofuses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replacethe core of that paper, which relies on non-positive mean curvature at singularpoints, by an argument based on scaling discrepancies, which promises to beapplicable to more general free boundary or free discontinuity problems.

A parabolic free boundary problem with Bernoulli type condition on the free boundary