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arXiv · 2609.14981

Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem

Farhan Abedin·William M. Feldman·Kerrek Stinson
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Abstract

We study the Perron extremal solutions of the stationary one-phase Bernoulli problem. These solutions are important in several applications, but not much is known about the regularity of their free boundaries due to their non-variational construction. We show that, in fact, Perron extremal solutions retain some variational features; specifically, they are solutions in the sense of inner variations. We establish this property by showing that extremal solutions arise as infinite-time limits of monotone solutions of the parabolic Bernoulli problem. As a consequence, we are able to describe the structure of singularities for Perron extremal solutions in two dimensions: largest subsolutions have smooth free boundaries, while smallest supersolutions have free boundary points that are either smooth, or whose blow-up limits are two-plane wedges with equal slope.

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Farhan Abedin, William M. Feldman, Kerrek Stinson. 2026-09-14. Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem. https://arxiv.org/abs/2609.14981

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