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On the convergence of stable phase transitions
Author(s) -
Padilla Pablo,
Tonegawa Yoshihiro
Publication year - 1998
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199806)51:6<551::aid-cpa1>3.0.co;2-6
Subject(s) - mathematics , convergence (economics) , domain (mathematical analysis) , sense (electronics) , stability (learning theory) , phase transition , energy (signal processing) , mathematical analysis , combinatorics , pure mathematics , statistics , thermodynamics , physics , economics , economic growth , engineering , machine learning , computer science , electrical engineering
We consider the local behavior of critical points of the functional $\int_U{\varepsilon |\nabla u^\varepsilon|^2} + {W(u^{\varepsilon})\over\varepsilon}\, dx$ as ε → 0. Here, W is a double‐well potential and U is a regular domain in ℝ n , n ≥ 2. Assuming that { u ε } 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.