On the covergence of minimizers of singular perturbation functionals

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 ≥ 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.

Hölder regularity and convergence for a non-local model of nematic liquid crystals in the large-domain limit

“…Up to a scaling, the statement given here is essentially the same as [10, Lemma B.2]. However, in [10] the potential is assumed to be finite and smooth on the whole of R m , while our potential ψ b is singular out of Q. Nevertheless, the proof carries over to our setting.…”

“…Studying the convergence of minimisers of Landau-de Gennes towards the Oseen-Frank limit has attracted interest, with Majumdar and Zarnescu showing global W 1,2 convergence and uniform convergence away from singular sets in the one-constant case [35], Nguyen and Zarnescu proving convergence results in stronger topologies [37], Contreras, Lamy and Rodiac generalising the approach to other harmonic-map problems [10], and further extensions by Contreras and Lamy [9] and Canevari, Majumdar and Stroffolini [7] to more general elastic energies. In other settings, the W 1,2 -convergence does not hold globally but only locally, away from the singular sets, due to topological obstructions carried by the boundary data and/or the domain (see e.g.…”

“…Nevertheless, the proof carries over to our setting. Indeed, the map w constructed in [10] takes values in a neighbourhood of N , whose thickness can be made arbitrarily small by choosing η small (see also [33,Lemma 1]). In particular, we can make sure that the image of w is contained in the set Q 0 , where the function ψ b is finite and smooth, and the arguments of [10] carry over.…”

“…Indeed, the map w constructed in [10] takes values in a neighbourhood of N , whose thickness can be made arbitrarily small by choosing η small (see also [33,Lemma 1]). In particular, we can make sure that the image of w is contained in the set Q 0 , where the function ψ b is finite and smooth, and the arguments of [10] carry over. Incidentally, Lemma 4.3 crucially depends on the non-degeneracy assumption (H 6 ) for the bulk potential ψ b .…”

Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

“…The Landau-de Gennes energy enforces this uniaxial constraint as a small coherence length goes to zero, the limit in which one can recover the Oseen-Frank model. This convergence has recently produced a rich trove of mathematical analysis [21,7,8,6,14,10]. An important feature in experiment and in the analysis is the occurrence of defects (singular structures).…”