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Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells

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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.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Fanghua Lin

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Changyou Wang

Authors
  1. Fanghua Lin
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  2. Changyou Wang
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Corresponding authors

Correspondence to Fanghua Lin or Changyou Wang.

Additional information

Dedicated to Professor Andrew J. Majda with deep admiration

This work was supported by NSF Grants DMS-1501000, DMS-1764417.

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Lin, F., Wang, C. Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells. Chin. Ann. Math. Ser. B 40, 781–810 (2019). https://doi.org/10.1007/s11401-019-0160-6

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  • DOI: https://doi.org/10.1007/s11401-019-0160-6

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