“The De Giorgi Conjecture for the Free Boundary Allen–Cahn Equation”
Venerdì 24 Aprile 2026, ore 15:30 - Aula 1AD100 - Alberto Figalli (ETH Zürich)
Abstract
The Allen–Cahn equation is well known as a diffuse-interface model that approximates minimal surfaces. This connection led to the conjecture that, in dimensions up to seven, all global stable solutions of the Allen–Cahn equation are one-dimensional. If true, this would in particular imply the celebrated De Giorgi conjecture for monotone solutions.
Motivated by the geometric character of the equation, David Jerison has long advocated that a free-boundary formulation offers a more natural framework for capturing its relationship with minimal surfaces. From this viewpoint, one is led to revisit the conjecture in a free-boundary setting.
In recent joint work with Chan, Fernández-Real, and Serra, we classify all global stable solutions to the one-phase Bernoulli free-boundary problem in three dimensions. As a consequence, we prove that global stable solutions to the free-boundary Allen–Cahn equation in three dimensions are necessarily one-dimensional.
