The Allen-Cahn (AC) equation is known to approximate minimal surfaces, leading to the conjecture that global stable solutions to AC should be one-dimensional in dimensions up to 7. If true, this result would imply the celebrated De Giorgi conjecture for monotone solutions.

Recognizing the interactive nature of the AC equation, Jerison has advocated for more than a decade that a free-boundary version of AC would offer a more natural framework for approximating minimal surfaces. This perspective motivates studying the above conjecture within this free-boundary context.

In recent joint work with Chan, Fernandez-Real, and Serra, we classify all stable global solutions to the one-phase Bernoulli free-boundary problem in three dimensions and, as a consequence, we establish that global stable 3d solutions to the free-boundary AC equation are one-dimensional.