Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem

Energy functionals describing phase transitions in crystalline solids are often non-quasiconvex and minimizers might therefore not exist. On the other hand, there might be infinitely many gradient Young measures, modelling microstructures, generated by minimizing sequences, and it is an open problem how to select the physical ones. In this work we consider the problem of selecting minimizing sequences for a one-dimensional three-well problem ε. We introduce a regularization εε of ε with an ε-small penalization of the second derivatives, and we obtain as ε ↓ 0 its Γ-limit and, under some further assumptions, the Γ-limit of a suitably rescaled version of εε. The latter selects a unique minimizing gradient Young measure of the former, which is supported just in two wells and not in three. We then show that some assumptions are necessary to derive the Γ-limit of the rescaled functional, but not to prove that minimizers of εε generate, as ε ↓ 0, Young measures supported just in two wells and not in three.

Rank-one convexity implies quasi-convexity on certain hypersurfaces

We show that, if f : M2×2 → R is rank-one convex on the hyperboloid is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on M2×2 uniformly on compact subsets of . Equivalently, every gradient Young measure supported on a compact subset of is a laminate.