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

Abstract: 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 is known about $\textit{critical points}$ 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 $u\text{---}$which can be obtained as limits of classical solutions or limits of a singular perturbation problem$\text{---}$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 $\textit{frequency formula}$ for the Bernoulli problem as well as the celebrated $\textit{Naber-Valtorta procedure}$ to answer this more than 20 year old question in an affirmative way:

For a closed class we call $\textit{variational solutions}$ of the Bernoulli problem, we show that the topological free boundary $\partial \{u > 0\}$ (including $\textit{degenerate}$ singular points $x$, at which $u(x + r \cdot)/r \rightarrow 0$ 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).

analysis of PDEsdifferential geometry

Audience: researchers in the topic

( paper )

---

NCTS international Geometric Measure Theory seminar

Series comments: We envisage an event built around virtual presentations on progress in geometric measure theory by external speakers. Every researcher is free to register as a participant and thus gain access to a virtual facility which is complete with lobby, lecture hall, and areas with boards for discussion. Thus, it shall recreate the exchange possibilities found at international conferences.

Focus: regularity and singularity theories for submanifolds of Riemannian manifolds and some of its applications.

Frequency: one presentation every other month.

Registration: required for new participants, go to the seminar website (allow at least one working day for processing).

Virtual venue: HyHyve space NCTS iGMT seminar (only for registered participants, opened one hour before the events).

You might want to consult the description of the premises and instructions.

Former organiser: Guido De Philippis (till March 2022).

---