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.
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

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.

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