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- W4221135616 abstract "<abstract><p>In this paper we apply a smoothing technique for the maximum function, based on the compensated convex transforms, originally proposed by Zhang in <sup>[<xref ref-type=bibr rid=b1>1</xref>]</sup> to construct some computable multiwell non-negative quasiconvex functions in the calculus of variations. Let $ Ksubseteq Esubseteq M^{mtimes n} $ with $ K $ a finite set in a linear subspace $ E $ without rank-one matrices of the space $ M^{mtimes n} $ of real $ mtimes n $ matrices. Our main aim is to construct computable quasiconvex lower bounds for the following two multiwell models with possibly uneven wells:</p> <p>i) Let $ f:Ksubseteq Eto E^perp $ be an $ L $-Lipschitz mapping with $ 0leq Lleq 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</p> <p>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_iin E $ with $ U_i:Eto 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.</p></abstract>" @default.
- W4221135616 created "2022-04-03" @default.
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- W4221135616 date "2022-01-01" @default.
- W4221135616 modified "2023-09-26" @default.
- W4221135616 title "Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices" @default.
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- W4221135616 doi "https://doi.org/10.3934/era.2022082" @default.
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