Refined approximation for a class of Landau-de Gennes energy minimizers

TL;DR

This paper refines the understanding of how Landau-de Gennes energy minimizers approximate Oseen-Frank minimizers as the elastic constant approaches zero, providing precise asymptotic expansions and convergence rates.

Contribution

It offers a sharper asymptotic analysis of Landau-de Gennes minimizers, identifying the first order correction and the equations governing it in the small elastic constant limit.

Findings

Established the functional setting for asymptotic expansion.

Derived the sharp rate of convergence as elastic constant tends to zero.

Formulated the equations for the first order correction term.

Abstract

We study a class of Landau-de Gennes energy functionals in the asymptotic regime of small elastic constant $L>0$. We revisit and sharpen the results in [18] on the convergence to the limit Oseen-Frank functional. We examine how the Landau-de Gennes global minimizers are approximated by the Oseen-Frank ones by determining the first order term in their asymptotic expansion as $L\to 0$. We identify the appropriate functional setting in which the asymptotic expansion holds, the sharp rate of convergence to the limit and determine the equation for the first order term. We find that the equation has a ``normal component'' given by an algebraic relation and a ``tangential component'' given by a linear system.