  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.
