On the Symmetric Lamination Convex and Quasiconvex Hull for the Coplanar $n$-Well Problem in Two Dimensions

Abstract

We study some particular cases of the $n$-well problem in two-dimensional linear elasticity. Assuming that every well in $\mathcal{U}\subset \mathbb{R}^{2\times 2}_{\text{sym}}$ belong to the same two-dimensional affine subspace, we characterize the symmetric lamination convex hull $L^{e}(\mathcal{U})$ for any number of wells in terms of the symmetric lamination convex hull of all three-well subsets contained in $\mathcal{U}$. For a family of four-well sets where two pairs of wells are rank-one compatible, we show that the symmetric lamination convex and quasiconvex hulls coincide, but are strictly contained in its convex hull $C(\mathcal{U})$. We extend this result to some particular configurations of $n$ wells. Most of the proofs are constructive, and we also present explicit examples.