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Local minimizers and slow motion for the mass preserving Allen-Cahn equation in higher dimensions


Authors: Giovanni Leoni and Ryan Murray
Journal: Proc. Amer. Math. Soc. 147 (2019), 5167-5182
MSC (2010): Primary 49J45, 35K10, 35K25
DOI: https://doi.org/10.1090/proc/13988
Published electronically: September 23, 2019
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Abstract: This paper completely resolves the asymptotic development of order $ 2$ by $ \Gamma $-convergence of the mass-constrained Cahn-Hilliard functional. Important new results on the slow motion of interfaces for the mass preserving Allen-Cahn equation and the Cahn-Hilliard equations in higher dimension are obtained as an application.


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Additional Information

Giovanni Leoni
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Ryan Murray
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695

DOI: https://doi.org/10.1090/proc/13988
Keywords: Second-order $\Gamma$-convergence, rearrangement, Cahn--Hilliard functional, slow motion, Allen--Cahn equation, Cah--Hilliard equation
Received by editor(s): July 15, 2017
Received by editor(s) in revised form: October 14, 2017
Published electronically: September 23, 2019
Additional Notes: The authors warmly thank the Center for Nonlinear Analysis, where part of this work was carried out. The center is partially supported by NSF Grant No. DMS-0635983 and NSF PIRE Grant No. OISE-0967140.
The research of the first author was partially funded by the NSF under Grants No. DMS-1412095 and DMS-1714098.
The research of the second author was supported by NSF PIRE Grant No. OISE-0967140.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2019 American Mathematical Society