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

Abstract

Consider the parabolic free boundary problem $$\Delta u-{\partial }_{t}u=0\text{in}\{u>0\},|\nabla u|=1\text{on}\partial \{u>0\}.$$ For a realistic class of solutions, containing for example {\em all} limits of the singular perturbation problem $$\Delta u_{\epsilon }-{\partial }_t{u}_{\epsilon }={\beta }_{\epsilon }(u_{\epsilon })\text{as}\epsilon \to 0,$$ 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 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.