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 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.