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

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

Rectifiability, finite Hausdorff measure, and compactness for non-minimizing Bernoulli free boundaries

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time-dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points $u$ -- which can be obtained as limits of classical solutions or limits of a singular perturbation problem -- it has been open since [Weiss03] whether the singular set can be large and what equation the measure $\Delta u$ satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber-Valtorta procedure to answer this more than 20 year old question in an affirmative way: For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary $\partial \{u > 0\}$ (including degenerate singular points $x$, at which $u(x + r \cdot)/r \rightarrow 0$ as $r\rightarrow 0$) is countably $\mathcal{H}^{n-1}$-rectifiable and has locally finite $\mathcal{H}^{n-1}$-measure, and we identify the measure $\Delta u$ completely. This gives a more precise characterization of the free boundary of $u$ in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them

The two-phase membrane problem — regularity of the free boundaries in higher dimensions

For the two-phase membrane problem \Delta u=\lambda_{+}\chi_{\{u>0\}}-\lambda_{-}\chi_{\{u<0\}}, where \lambda_{+} and \lambda_{-} are positive Lipschitz functions, we prove in higher dimensions that the free boundary is in a neighborhood of each "branch point\u27; the union of two C^{1}-graphs. The result is optimal in the sense that these graphs are in general not of class C^{1,\mbox{Dini}}, as shown by a counter-example. As application we obtain a stability result with respect to perturbations of the boundary data