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

Maggi, Francesco; Restrepo, Daniel

doi:10.2140/apde.2024.17.1761

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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

DOI: 10.2140/apde.2024.17.1761

Abstract

We consider the isoperimetric problem defined on the whole $ℝ^{n}$ 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.

Keywords

isoperimetric problem, Allen–Cahn energy, Alexandrov theorem

Mathematical Subject Classification

Primary: 35B35, 35J91

Secondary: 35J93, 49Q20

Milestones

Received: 23 February 2022

Revised: 5 October 2022

Accepted: 8 November 2022

Published: 20 June 2024

Authors

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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Vol. 17, No. 5, 2024