An important question in the calculus of variations is whether a given energy functional has a minimizer over some class of admissible functions. The direct method reduces this to the question of whether the energy functional is lower semi-continuous over this class. Thanks to the work of Murat-Tartar ’79, Fonseca-Müller ’99 and more recently Guerra-Raiță ’20, we can determine the ‘correct’ condition on the energy functional for lower semi-continuity, whenever the class of functions satisfies a prescribed PDE constraint. In this talk I will discuss these results, and we will see how the presence of the differential constraint forces oscillation/concentration phenomena in sequences to be restricted to only certain directions.