2017/12/01 by Dmitry Golovaty, Golovaty, Dmitry, Peter Sternberg +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1712.00493
openalex publication_date 2017/12/01 · openalex created_date 2022/09/26 · openalex updated_date 2026/07/28
We carry out an asymptotic analysis of a thin nematic liquid crystal in which\none elastic constant dominates over the others, namely n
labelenergyab
inf E_
varepsilon(u)
quad
mboxwhere
quad E_
varepsilon(u)\n:=
frac12
int_
Omega
left
varepsilon
,|
nabla u|2 +\n
frac1
varepsilon
,(|u|2 - 1)2 + L
,(
mathrmdiv
,u)2
right
,dx.\n Here u: \Ω \→ mathbb R2 is a vector field, 0 <\n\ε \≪ 1 is a small parameter, and L > 0 is a fixed constant,\nindependent of \ε. We derive the \Γ-limit E0, which is a\nsum of a bulk term penalizing divergence and an Aviles-Giga type wall energy\ninvolving the cube of the jump in the tangential component of the\n mathbbS1-valued order parameter. We then derive criticality conditions\nfor E0 and analyze minimization of E0 both rigorously and numerically for\nvarious domains \Ω and a variety of Dirichlet boundary conditions.\n