Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices

Zhang, Kewei; Yin, Ke

Abstract

In this paper we apply a smoothing technique for the maximum function, based on the compensated convex transforms, originally proposed by Zhang in [1] to construct some computable multiwell non-negative quasiconvex functions in the calculus of variations. Let K ⊆ E ⊆ Mm×n with K a finite set in a linear subspace E without rank-one matrices of the space Mm×n of real m × n matrices. Our main aim is to construct computable quasiconvex lower bounds for the following two multiwell models with possibly uneven wells: i) Let f: K ⊆ E → E⊥ be an L-Lipschitz mapping with 0 ≤ L ≤ 1/α and H2(X) = min{|PEX − Ai|2 + α|PE⊥X − f (Ai)|2 + βi: i = 1, 2,…, k}, where α > 0 is a control parameter, and ii)(Formula Presented), where Ai ∈ E with Ui: E → E invertible linear transforms for i = 1, 2,…, k. If the control paramenter α > 0 is sufficiently large, our quasiconvex lower bounds are ‘tight’ in the sense that near each ‘well’ the lower bound agrees with the original function, and our lower bound are of C1,1. We also consider generalisations of our constructions to other simple geometrical multiwell models and discuss the implications of our constructions to the corresponding variational problems

Citation

Zhang, K., & Yin, K. (2022). Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices. Electronic Research Archive, 30(5), 1632-1652. https://doi.org/10.3934/era.2022082

• Journal Article Type: Article
• Acceptance Date: Feb 17, 2022
• Online Publication Date: Mar 24, 2022
• Publication Date: 2022
• Deposit Date: Mar 10, 2022
• Publicly Available Date: Mar 24, 2022
• Journal: Electronic Research Archive
• Electronic ISSN: 2688-1594
• Peer Reviewed: Peer Reviewed
• Volume: 30
• Issue: 5
• Pages: 1632-1652
• DOI: https://doi.org/10.3934/era.2022082
• Keywords: multiwell models, vectorial calculus of variations, quasiconvex functions, quasicon- vex envelope, quasiconvex lower bounds, computational lower boundes, translation method, maximum function, compensated convex transforms, C1;1-smooth approximation
• Public URL: https://nottingham-repository.worktribe.com/output/7569907
• Publisher URL: http://www.aimspress.com/article/doi/10.3934/era.2022082
• Additional Information: This is a joint work with Ke Yin