Consider the parabolic free boundary problem Δu−∂t​u=0in{u>0},∣∇u∣=1on∂{u>0}. For a
realistic class of solutions, containing for example {\em all} limits of the
singular perturbation problem Δuϵ​−∂t​uϵ​=βϵ​(uϵ​)asϵ→0, we prove that one-sided
flatness of the free boundary implies regularity.
In particular, we show that the topological free boundary ∂{u>0}
can be decomposed into an {\em open} regular set (relative to
∂{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