Boundary regularity of weakly anchored harmonic maps

TL;DR

This paper proves optimal boundary regularity for minimizers of weak anchoring energies in liquid crystal models across all dimensions, revealing full boundary regularity in three dimensions and linking regularity to limit problems.

Contribution

It establishes the first comprehensive boundary regularity results for weakly anchored harmonic maps, especially in three dimensions, contrasting physics observations.

Findings

Optimal boundary regularity in all dimensions n≥3

Full boundary regularity in 3D contrasts with boundary defects in physics

Regularity inheritance from limit problems for weak and strong anchoring cases

Abstract

In this note we study the boundary regularity of minimizers of a family of weak anchoring energies that model the states of liquid crystals. We establish optimal boundary regularity in all dimensions $n\geq 3 .$ In dimension $n=3,$ this yields full regularity at the boundary which stands in sharp contrast with the observation of boundary defects in physics works. We also show that, in the cases of weak and strong anchoring, regularity of minimizers is inherited from that of their corresponding limit problems.The analysis rests in a crucial manner on the fact that the surface and Dirichlet energies scale differently; we take advantage of this fact to reduce the problem to the known regularity of tangent maps with zero Neumann conditions.