A basic difficulty in the search for minimizers over subsets of $W^{1,p}(\Omega;\mathbb{R}^D)$ is that weakly convergent sequences may develop increasingly fine oscillations that affect nonlinear energies of the form $I[u] = \int_{\Omega} F(x, u(x), \nabla u(x)) dx$.

In this introductory talk, I will review classical ideas originating in the work of Charles Morrey and explain, at a conceptual level, how quasiconvexification and gradient Young measures capture such oscillations and lead to relaxed variational problems for which existence can be recovered. I will then turn to variational problems involving noncommuting variables, where a different obstruction may occur : a minimizing sequence may converge at the level of noncommutative laws while the limiting law cannot be realized in the prescribed ambient algebra. I will illustrate this phenomenon through elementary examples related to noncommutative optimal transport and discuss a corresponding notion of relaxation obtained by enlarging the ambient algebra.