### Abstract

This is in the sequel of authors’ paper [Lin, F. H., Pan, X. B. and Wang, C. Y., Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6), 2012, 833-888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition. The authors also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg’s work (in 1989) on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.

Original language | English (US) |
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Pages (from-to) | 781-810 |

Number of pages | 30 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 40 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2019 |

### Keywords

- 35J50
- Boundary monotonicity inequality
- Boundary partial regularity
- Partially free and partially constrained boundary

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics