Some Results on quasiconvex hull for a n-well problem in 2D under Geometrically Linear Elastic regime

In shape-memory alloys, it is common to analyze pattern formation induced by the existence of different zero free energy phases in the material. These microstructure provokes the shape-memory effect. The classical model used to study the problem is \begin{equation}\label{1} \inf_{\substack{y\in W^{1,\infty}(\Omega)\\ y|_{\partial \Omega}= Fx}} \ \int_\Omega \phi\left(\nabla y\right) \,dx, \quad %\begin{array}{l} \ker\, \phi = \{u_1,\, u_2,\, \dots,\, u_n\}\, \oplus\, \mathbb{R}^{2\times 2}_{skew},%\\ \{u_1,\, u_2,\, u_3\} \subset \mathbb{R}^{2\times 2}_{sym} \end{array} \end{equation} where $\{u_1,\, u_2,\, \dots,\, u_n\} \subset \mathbb{R}^{2\times 2}_{sym}$ and the function $\phi:\mathbb{R}^{2\times 2}\rightarrow \mathbb{R}$ satisfies mild growth conditions and it is invariant under addition of skew-symmetric matrices to its argument.

In this talk, I will present some recent results about the relaxed problem via quasiconvexification. More precisely, it will be proved that the quasiconvex hull of $\ker \phi$ equals its convex hull if the $n$ wells are pairsewise symmetrized-rank-one connected. Particularly, in the three-well problem if one of this connection is lost, then the contention of the quasiconvex hull of $\ker \phi$ in its convex hull is strict.