Perron Extremal Solutions of the Bernoulli One-Phase Problem
Speaker: Will FeldmanDate of Talk: October 23, 2026
Upstream link: Joint Caltech/UCLA Analysis Seminar
1. The Setting
The Bernoulli one-phase problem asks for a function \(u : U \to [0, \infty)\) satisfying
\[\begin{cases} \Delta u = 0 & \textrm{on } \left\lbrace u > 0 \right\rbrace, \\ \left\lvert \nabla u \right\rvert = Q(x) & \textrm{on } \partial \left\lbrace u > 0 \right\rbrace. \end{cases}\]Typically one takes \(Q(x)\equiv 1\). There are several common notions of solution to this, each with some kind of nonuniqueness, such as
- classical solutions,
- minimisers of the Alt-Caffarelli functional (perhaps with stability, or minimising in a subset of competitors),
- variational solutions (a certain critical point of the functional),
- viscosity solutions, with distinguished solutions such as the minimal supersolution of Perron.
2. An Energetic Connection to Perron Solutions
There are rather complicated relationships between these classes, and it can be hard to distinguish between the Perron extremal solutions and the energy minimiser(s). But there is a key idea that characterises these Perron solutions as limits of the parabolic flows with the appropriate boundary data!
Specifically, if \(g_0\) is a subsolution, one can take the parabolic Bernoulli flow starting from \(g_0\); over time, it increases in energy and tends to a limit, and this limit is the maximal subsolution. Analogously for the minimal supersolution.
(Actually proving this is rather difficult, though, and one has to first prove this for an approximate parabolic Bernoulli problem first and then send a parameter \(\epsilon \to 0\). )
3. Some Special Viscosity Solutions
When \(0 \leq \alpha \leq 1\), the function \(u(t, x) = \alpha \left\lvert x_1 \right\rvert\) e.g. is a viscosity supersolution to the one-phase Bernoulli problem but very clearly is not a classical solution. However, fluid dynamicists Baker, Saffman, Sheffield in 1976 and Traizet in 2014 gave examples of singly periodic classical solutions forming a line of “hollow vortices” that approach \(\alpha \left\lvert x_1 \right\rvert\) upon zooming out, i.e. these viscosity supersolutions are limits of classical solutions. See also the so-called “hairpin solution”!