Abstract
We study small energy solutions within the Landau-de Gennes theory for nematic liquid crystals, subject to Dirichlet boundary conditions. We consider two-dimensional and three-dimensional domains separately. In the two-dimensional case, we establish the equivalence of the Landau-de Gennes and Ginzburg-Landau theory. In the three-dimensional case, we give a new definition of the defect set based on the normalized energy. In the threedimensional uniaxial case, we demonstrate the equivalence between the defect set and the isotropic set and prove the C 1,α-convergence of uniaxial small energy solutions to a limiting harmonic map, away from the defect set, for some 0 < a < 1, in the vanishing core limit. Generalizations for biaxial small energy solutions are also discussed, which include physically relevant estimates for the solution and its scalar order parameters. This work is motivated by the study of defects in liquid crystalline systems and their applications.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1303-1337 |
| Number of pages | 35 |
| Journal | Communications on Pure and Applied Analysis |
| Volume | 11 |
| Issue number | 3 |
| DOIs | |
| State | Published - Jun 22 2012 |
| Externally published | Yes |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: A. Majumdar is supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST), to the Oxford Centre for Collaborative Applied Mathematics and an EPSRC Career Acceleration Fellowship EP/J001686/1.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.