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 $$\text{Missing end{equation}}$$ where $\{u_1,u_2,\dots ,u_n\}\subset {\mathbb{R}}_{sym}^{2\times 2}$ and the function $\varphi :{\mathbb{R}}^{2\times 2}\to \mathbb{R}$ 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 $\ker \varphi$ 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 $\ker \varphi$ in its convex hull is strict.