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Uniform stability in the Euclidean isoperimetric problem for the Allen–Cahn energy

Francesco Maggi and Daniel Restrepo

Vol. 17 (2024), No. 5, 1761–1830

We consider the isoperimetric problem defined on the whole n by the Allen–Cahn energy functional. For nondegenerate double-well potentials, we prove sharp quantitative stability inequalities of quadratic type which are uniform in the length scale of the phase transitions. We also derive a rigidity theorem for critical points analogous to the classical Alexandrov theorem for constant mean curvature boundaries.

isoperimetric problem, Allen–Cahn energy, Alexandrov theorem
Mathematical Subject Classification
Primary: 35B35, 35J91
Secondary: 35J93, 49Q20
Received: 23 February 2022
Revised: 5 October 2022
Accepted: 8 November 2022
Published: 20 June 2024
Francesco Maggi
Department of Mathematics
The University of Texas at Austin
Austin, TX
United States
Daniel Restrepo
Department of Mathematics
The University of Texas at Austin
Austin, TX
United States

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