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

Ke Yin; Kewei Zhang; Ke Yin; Kewei Zhang

doi:10.3934/era.2022082

Electronic Research Archive

2022, Volume 30, Issue 5: 1632-1652. doi: 10.3934/era.2022082

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

Ke Yin ^{1
,
,},
Kewei Zhang ^{2
,
,}

1.
Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China
2.
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Received: 22 August 2021 Revised: 12 February 2022 Accepted: 17 February 2022 Published: 24 March 2022

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 $K\subseteq E\subseteq M^{m\times n}$ with $K$ a finite set in a linear subspace $E$ without rank-one matrices of the space $M^{m\times n}$ of real $m\times n$ matrices. Our main aim is to construct computable quasiconvex lower bounds for the following two multiwell models with possibly uneven wells:

i) Let $f:K\subseteq E\to E^\perp$ be an $L$ -Lipschitz mapping with $0\leq L\leq 1/\alpha$ and $H_2(X) = \min\{ |P_EX-A_i|^2+\alpha|P_{E^\perp}X-f(A_i)|^2+\beta_i:\, i = 1, 2, \dots, k\}$ , where $\alpha > 0$ is a control parameter, and

ii) $H_1(X) = \alpha|P_{E^\perp}X|^2+\min\{\sqrt{|\mathcal{U}_i(P_EX-A_i)|^2+\gamma_i}: i = 1, 2, \dots, k\}$ , where $A_i\in E$ with $U_i:E\to E$ invertible linear transforms for $i = 1, 2, \dots, k$ . If the control paramenter $\alpha > 0$ 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 $C^{1, 1}$ . 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.

Keywords:

Citation: Ke Yin, Kewei Zhang. Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices[J]. Electronic Research Archive, 2022, 30(5): 1632-1652. doi: 10.3934/era.2022082

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Abstract

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