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

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 multip...

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Bibliographic Details
Published in:Chinese annals of mathematics. Serie B 2019-09, Vol.40 (5), p.781-810
Main Authors: Lin, Fanghua, Wang, Changyou
Format: Article
Language:English
Subjects:
Applications of Mathematics
Boundary conditions
Curvature
Diffusion rate
Gradient flow
Mathematics
Mathematics and Statistics
Phase transitions
Wells
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Summary: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.
ISSN:0252-9599
1860-6261
DOI:10.1007/s11401-019-0160-6