HyHyve, Online seminar

(線上演講 HyHyve)

Rectifiability, Finite Hausdorff Measure, and Compactness for Non-minimizing Bernoulli Free Boundaries

Georg Weiss (Universität Duisburg-Essen)

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 *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 ∆*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 ∂{*u* > 0} (including *degenerate* singular points *x*, at which *u*(*x* + *r*·)/*r* → 0 as *r* → 0) is countably*H*^{n}^{-1}- rectifiable and has locally finite *H*^{n}^{−1}-measure, and we identify the measure ∆*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 wellas 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).

Agenda

Get-together (30 min)

Presentation Georg Weiss (60 min)

Questions and discussions (30 min)

Seminar website https://ncts.ntu.edu.tw/gmt-seminar.html