Integrability of known singular one-phase cones

Determine whether the one-phase cones constructed by Hong, Wang and Hong, Wang, and Hines, Kolesar and McGrath are integrable in the sense that every 1-homogeneous Jacobi field is generated by a family of 1-homogeneous classical one-phase solutions.

Background

The paper studies uniqueness of blow-ups for variational solutions of the one-phase free boundary problem when a blow-up is a one-phase cone with an isolated singularity. Its main theorem proves uniqueness without assuming that the cone is integrable, although integrability yields stronger, polynomial convergence rates.

Several families of singular one-phase cones have been constructed in dimensions three and higher, including axially symmetric, Lawson-type, isoparametric, and discrete-group-invariant examples. The authors note that the structural property of integrability has not been established for the cones constructed in the cited works, leaving open whether the stronger convergence theory applies to them.