[CvGmt News] [cvgmt seminar] The De Giorgi Conjecture for the Free Boundary Allen–Cahn Equation

[Esther.Cabezas-Rivas at uv.es]

The De Giorgi Conjecture for the Free Boundary Allen–Cahn Equation
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http://cvgmt.sns.it/seminar/978/

Date: Monday, Mar 30, 2026

Time: 11:00

Speaker: Alessio Figalli

BANG (Bridging ANalysis & Geometry) online seminar

organized by Esther Cabezas-Rivas & Salvador Moll (Universitat de València)

Abstract. The Allen–Cahn equation is known to approximate minimal surfaces. This connection led to the conjecture that global stable solutions of the Allen–Cahn equation should be one-dimensional in dimensions up to seven. If true, this statement would imply the celebrated De Giorgi conjecture for monotone solutions.

Motivated by the geometric nature of the Allen–Cahn equation, Jerison has advocated for more than a decade that a free-boundary formulation provides a more natural framework for approximating minimal surfaces. This perspective leads to studying the above conjecture in the free-boundary setting.

In recent joint work with Chan, Fernández-Real, and Serra, we classify all stable global solutions to the one-phase Bernoulli free-boundary problem in three dimensions. As a consequence, we show that global stable solutions to the free-boundary Allen–Cahn equation in three dimensions are one-dimensional.

Register to receive a zoom link: https://sites.google.com/view/bangseminar/bang