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 $u(x + r)/r \to 0$ as $r \to 0$) 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.
This is a joint work with Dennis Kriventsov (Rutgers).
http://cvgmt.sns.it/seminar/885/
When | Wed Sep 20, 2023 8am – 9am Coordinated Universal Time |