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

@misc{abedin2026variational,
  title = {{Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem}},
  author = {Farhan Abedin and William M. Feldman and Kerrek Stinson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.14981},
  eprint = {2609.14981},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14981},
}

TY  - UNPB
TI  - Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem
AU  - Farhan Abedin
AU  - William M. Feldman
AU  - Kerrek Stinson
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.14981
UR  - https://arxiv.org/abs/2609.14981
AB  - 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.
ER  -