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

Abstract

We consider the isoperimetric problem defined on the whole ${\mathbb{R}}^n$ by the Allen--Cahn energy functional. For non-degenerate 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's theorem for constant mean curvature boundaries.

61 pages

References in corpus (2)

• Alexandrov's theorem revisited
• Second-Order $\Gamma$-limit for the Cahn-Hilliard Functional