On the convergence of stable phase transitions

Abstract

We consider the local behavior of critical points of the functional as ε → 0. Here, W is a double-well potential and U is a regular domain in ℝn, n ≥ 2. Assuming that {uε}ε>0 is stable for n = 2 and locally energy-minimizing for n = 3, we show that the level sets of solutions converge in an average sense to a stationary (n − 1)-rectifiable varifold. Our study is based on estimates derived from the second variation formula and is entirely local. © 1998 John Wiley & Sons, Inc.