On the convergence of minimizers of singular perturbation functionals

TL;DR

This paper proves the convergence of minimizers of singular perturbation functionals to harmonic maps in various dimensions, extending previous results to more general nonlinearities and boundary conditions.

Contribution

It generalizes convergence results for singular perturbation minimizers to all dimensions ≥3 and to broader nonlinearities without relying on specific potential forms.

Findings

Convergence of minimizers to harmonic maps is established under general assumptions.

Results extend to higher dimensions and more general nonlinearities.

Boundary convergence is uniform away from singular sets.

Abstract

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.