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 $0\le L\le 1/\alpha$ and $H_2(X)=min\{|P_{E}X-A_i{|}^2+\alpha |P_{E^{\perp }}X-f(A_i){|}^2+{\beta }_i:i=1,2,\dots ,k\}$, where $\alpha >0$ is a control parameter, and

ii) $H_1(X)=\alpha |P_{E^{\perp }}X{|}^2+min\{\sqrt{|{\mathcal{U}}_i(P_{E}X-A_i){|}^2+{\gamma }_i}:i=1,2,\dots ,k\}$, 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.