Boundary regularity for Oseen-Frank minimizers with arbitrary positive splay, twist, and bend constants

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be a smooth bounded domain and let $g:\partial \Omega \to {\mathbb{S}}^2$ be smooth. We prove that any global minimizer of the Oseen--Frank energy subject to the strong anchoring condition $n=g$ is smooth in a full neighborhood of $\partial \Omega$. The result holds for arbitrary positive splay, twist, and bend constants, without any small-anisotropy assumption. It resolves the boundary-regularity part of Lin and Liu's Problem~(c) for the pure Oseen--Frank problem and for its prescribed smooth magnetic-field perturbation.

31 pages, comments are welcome