The symmetric quasiconvex and lamination convex hull for the coplanar n-well problem and its relation to pattern formation in thin-film shape memory alloys

Ponente(s): Lauro Morales Montesinos, Antonio Capella Kort

In shape-memory alloys, it is common to analyze pattern formation induced by the existence of a finite number of zero-energy material phases U. These phases are characterized as minima of a non-convex bulk energy functional, I(U), which exhibits scale-invariance. This scale-invariance leads to the existence of minimizing sequences that weakly converge to constant phases. These constant weak limits, while not minima of I(U), are interpreted as "averaged" phases that produce the material's microstructure. The set of all such constant weak limits is referred to as the "quasiconvex hull", QU. Determining QU is a challenging task, and only a few examples of sets U have explicit sets QU. It is established that U is contained within QU, and QU is a subset of the convex hull, CU, of U. In this talk, we will investigate the quasiconvex hull QU for a finite set U made of 2x2 symmetric matrices, minima of an energy functional that depends on the symmetric part of its argument. We will identify an explicit set BU such that $QU \subseteq BU \subset CU$. Additionally, we will explore the conditions under which equality between QU and BU holds, while BU remains distinct from CU.