  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 $n\geq 3$, and to
general nonlinearities. Our proof follows a well-known scheme, relying on small
energy estimate and 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 3-dimensional settings. In higher dimensions we
also deal with additional difficulties concerning the boundary monotonicity
formula.
