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Transactions of the American Mathematical Society

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Existence of surface energy minimizing partitions of $ \mathbb{R}^{n}$ satisfying volume constraints


Author: David G. Caraballo
Journal: Trans. Amer. Math. Soc. 369 (2017), 1517-1546
MSC (2010): Primary 49Q20, 49J45
DOI: https://doi.org/10.1090/tran/6630
Published electronically: November 16, 2016
MathSciNet review: 3581211
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Abstract: We give the first proof, with independent smooth norms $ \phi _{ij},$ of the existence of surface energy minimizing partitions of $ \mathbb{R}^{n}$ into regions having prescribed volumes. Our existence proof significantly extends that of F. Almgren, who in 1976 gave the first such results for the special case in which each $ \phi _{ij}$ is a scalar multiple of a fixed smooth $ \phi :$ $ \phi _{ij}=c_{ij}\phi $. Most materials are polycrystalline and do not have surface energy density functions which are scalar multiples of one another, so it is important to extend the theory by removing this restriction, as we have done. We also discuss connections with polycrystalline evolution problems.


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Additional Information

David G. Caraballo
Affiliation: Department of Mathematics and Statistics, St. Mary’s Hall, 3rd floor, Georgetown University, Washington, DC 20057-1233
Email: dgc3@georgetown.edu

DOI: https://doi.org/10.1090/tran/6630
Keywords: Partition, polycrystal, cluster, Wulff crystal, surface energy minimizing, volume constraints, immiscible fluids
Received by editor(s): April 5, 2011
Received by editor(s) in revised form: June 29, 2014, and October 10, 2014
Published electronically: November 16, 2016
Dedicated: This paper is dedicated to the memory of my parents, Dulce M. and Marino Caraballo
Article copyright: © Copyright 2016 by the author