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Uniform bound on the number of partitions for optimal configurations of the Ohta–Kawasaki energy in 3D

Part of: Parabolic equations and systems Variational principles of physics

Published online by Cambridge University Press:  24 May 2023

Xin Yang Lu  and
Jun-cheng Wei
Xin Yang Lu*
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 1L1, Canada
Jun-cheng Wei
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada e-mail: jcwei@math.ubc.ca
*
e-mail: xlu8@lakeheadu.ca
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Abstract

We study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa–Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.

Keywords

Pattern formationsmall volume-fraction limittriblock copolymers

MSC classification

Primary: 49S05: Variational principles of physics (should also be assigned at least one other classification number in section 49)
Secondary: 35K30: Initial value problems for higher-order parabolic equations 35K55: Nonlinear parabolic equations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 4 , December 2023 , pp. 1341 - 1353
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

X.Y.L. acknowledges the support of the NSERC. The research of J.W. is partially supported by the NSERC.

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