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

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.