Singularity models for the Bernoulli free boundary problem from isoparametric hypersurfaces
Abstract
We develop a general construction of homogeneous solutions to the Bernoulli free boundary problem, as well as general extremal domains on the sphere, from isoparametric foliations of the sphere. Our construction produces rich families of infinitely many new examples with sophisticated topologies connected to minimal surfaces of the sphere by smooth families of interpolating capillary surfaces, which include novel singularity models in low dimensions and recover most known homogeneous one-phase solutions. We also introduce a geometric representation for every such homogeneous map and establish a geometric rigidity theorem: for every prescribed isoparametric foliation, and in particular for cohomogeneity-one subgroups of O(n), its radial geometry uniquely determines the corresponding homogeneous solution. Finally, we study the density of the constructed solutions and conjecture that the lowest density among non-flat cones in a given dimension n ≥ 5 is attained by an O(k) × O(n−k)-invariant cone.
Related
- On the Large Λ Asymptotics of the One Phase Bernoulli Free Boundary Problem
- Construction of compact embedded λ-hypersurfaces via isoparametric hypersurfaces
- Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
- Shape Optimization and Bernoulli Free Boundary Problems for the Riemannian p-Laplacian
- The stability conjecture for the Bernoulli problem in dimension four
BibTeX
@misc{firester2026singularity,
title = {{Singularity models for the Bernoulli free boundary problem from isoparametric hypersurfaces}},
author = {Benjy Firester and Raphael Tsiamis and Zihui Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31617},
eprint = {2609.31617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31617},
}
RIS
TY - UNPB
TI - Singularity models for the Bernoulli free boundary problem from isoparametric hypersurfaces
AU - Benjy Firester
AU - Raphael Tsiamis
AU - Zihui Zhao
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31617
UR - https://arxiv.org/abs/2609.31617
AB - We develop a general construction of homogeneous solutions to the Bernoulli free boundary problem, as well as general extremal domains on the sphere, from isoparametric foliations of the sphere. Our construction produces rich families of infinitely many new examples with sophisticated topologies connected to minimal surfaces of the sphere by smooth families of interpolating capillary surfaces, which include novel singularity models in low dimensions and recover most known homogeneous one-phase solutions. We also introduce a geometric representation for every such homogeneous map and establish a geometric rigidity theorem: for every prescribed isoparametric foliation, and in particular for cohomogeneity-one subgroups of O(n), its radial geometry uniquely determines the corresponding homogeneous solution. Finally, we study the density of the constructed solutions and conjecture that the lowest density among non-flat cones in a given dimension n ≥ 5 is attained by an O(k) × O(n−k)-invariant cone.
ER -