Nonlinear-regime subsolutions in two dimensions

Construct subsolutions for the cylindrical Bernoulli barrier problem in the nonlinear regime in dimension two, analogous to the nonlinear-regime subsolutions established in dimensions at least three, in order to complete the barrier construction for positive defect strengths.

Background

The paper constructs quantitative sub- and supersolution barriers for a cylindrical Bernoulli problem modeling a compactly supported defect. In the nonlinear regime, the supersolution construction works in every dimension, whereas the subsolution construction relies on a line-segment potential and is proved only for dimensions at least three.

The authors explain that the two-dimensional obstruction is connected with the logarithmic growth of the free boundary: a nontrivial capacity forces the free boundary to extend into the region below the planar profile, preventing the geometric property used in the higher-dimensional subsolution construction. Resolving this question would yield the missing nonlinear-regime subsolution theory in dimension two.

References

Whether nonlinear-regime subsolutions exist in $d=2$ remains an interesting open question.

— Solutions of the Bernoulli one-phase problem with a defect  (2609.17066 - Feldman et al., 15 Sep 2026) in Introduction, subsection “Main results”; Section 5, subsection “Barriers for large defects: subsolutions in d ≥ 3”

A well-known conjecture concerning stagnation points was formulated by G. Stokes in 1880 , which states that the interface between water and air forms a 120\circ opening angle at the stagnation point.

— A minimization problem for a cooperative system with a degenerate Bernoulli weight  (2609.25544 - Du et al., 22 Sep 2026) in Section 1, subsection “Introduction to the variational problem”