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 $\mathcal{E}$. We introduce a regularization $\mathcal{E}^\varepsilon$ of $\mathcal{E}$ with an $\varepsilon$-small penalization of the second derivatives, and we obtain as $\varepsilon\downarrow 0$ its $\Gamma-$limit and, under some further assumptions, the $\Gamma-$limit of a suitably rescaled version of $\mathcal{E}^\varepsilon$. 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 $\Gamma-$limit of the rescaled functional, but not to prove that minimizers of $\mathcal{E}^\varepsilon$ generate, as $\varepsilon\downarrow 0$, Young measures supported just in two wells and not in three.