The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work, we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower-dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions n >= 3, and to general nonlinearities. Our proof follows a well-known scheme, relying on a small energy estimate and a monotonicity formula. It departs substantially from previous studies in the treatment of the small energy estimate at the boundary, since we do not rely on the specific form of the potential. In particular, this extends existing results in three-dimensional settings. In higher dimensions, we also deal with additional difficulties concerning the boundary monotonicity formula.