Open AccessAn SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects
Author(s) -
Shankar C. Venkataramani
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2022101
Subject(s) - discretization , homogeneous space , relaxation (psychology) , bounded function , class (philosophy) , energy (signal processing) , character (mathematics) , gauge (firearms) , bounded variation , mathematics , statistical physics , computer science , topology (electrical circuits) , pure mathematics , mathematical analysis , physics , geometry , artificial intelligence , combinatorics , materials science , psychology , social psychology , statistics , metallurgy
We investigate stripe patterns formation far from threshold using a combination of topological, analytic, and numerical methods. We first give a definition of the mathematical structure of 'multi-valued' phase functions that are needed for describing layered structures or stripe patterns containing defects. This definition yields insight into the appropriate 'gauge symmetries' of patterns, and leads to the formulation of variational problems, in the class of special functions with bounded variation, to model patterns with defects. We then discuss approaches to discretize and numerically solve these variational problems. These energy minimizing solutions support defects having the same character as seen in experiments.