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

Consider the parabolic free boundary problem Δu – ∂ t u = 0 in {u > 0}, |∇u| = 1 on ∂{u > 0}. For a realistic class of solutions, containing for example all limits of the singular perturbation problem Δuε – ∂ t uε = βε (uε ) as ε → 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 open regular set (relative to ∂{u > 0}) which is locally a surface with Hölder-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