Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions

Jan Bouwe Van Den Berg, J. F. Williams

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In this paper we develop a symmetry preserving method for the rigorous computation of stationary states of the Ohta-Kawasaki partial differential equation in three space dimensions. By preserving the relevant symmetries we achieve an enormous reduction in computational cost. This makes it feasible to construct computer-assisted proofs of complex three-dimensional structures. In particular, we provide the first existence proofs for both the double gyroid and body centered cubic packed sphere solutions to this problem.

Original languageEnglish
Pages (from-to)131-158
Number of pages28
JournalSIAM Journal on Mathematical Analysis
Issue number1
Early online date29 Jan 2019
Publication statusPublished - Jan 2019


∗Received by the editors November 6, 2017; accepted for publication (in revised form) November 28, 2018; published electronically January 29, 2019. http://www.siam.org/journals/sima/51-1/M115562.html Funding: The work of the first author was partially supported by NWO-VICI grant 639033109. †VU Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands (janbouwe@few.vu.nl). ‡Simon Fraser University, Department of Mathematics, 8888 University Drive Burnaby, BC, V5A 1S6, Canada (jfwillia@sfu.ca).

FundersFunder number


    • 3D periodic structures
    • Computer-assisted proof
    • Otha-kawasaki problem
    • Partial differential equation
    • Symmetry preservation


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