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

We consider the isoperimetric problem defined on the whole 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.Contents 1. Introduction 1 2. Existence and radial decreasing symmetry of minimizers 8 3. Resolution of almost-minimizing sequences 20 4. Strict stability among radial functions 32 5. Proof of the uniform stability theorem 42 6.Proof of the Alexandrov-type theorem 51 Appendix A.