Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem

2019/11/29 by Jan Bouwe van den Berg, Berg, Jan Bouwe van den, Jim Williams +2 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #35B10 #35J60 #65G20 #74G65 #Analysis of PDEs (math.AP) #Diffusion and Search Dynamics #Dynamical Systems (math.DS) #FOS: Mathematics #Opinion Dynamics and Social Influence #Theoretical and Computational Physics #math.AP #math.DS #msc:35B10 #msc:35J60 #msc:65G20 #msc:74G65

paper · pdf · doi:10.48550/arxiv.1912.00059

40 pages, 9 figures

arxiv created 2019/11/29 · openalex publication_date 2019/11/29 · arxiv updated 2019/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of rigorously computing periodic minimizers to the Ohta-Kawasaki energy. We develop a method to prove existence of solutions and determine rigorous bounds on the distance between our numerical approximations and the true infinite dimensional solution and also on the energy. We use a method with prescribed symmetries to explore the phase space, computing candidate minimizers both with and without experimentally observed symmetries. We find qualitative differences between the phase diagram of the Ohta-Kawasaki energy and self consistent field theory when well away form the weak segregation limit.

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• Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
• Stability analysis for localized solutions in PDEs and nonlocal equations on ℝ^m

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