{
  "id": "2112.03453",
  "version": "v2",
  "published": "2021-12-07T02:29:45.000Z",
  "updated": "2022-09-28T23:26:22.000Z",
  "title": "Existence of minimizers and convergence of critical points for a new Landau-de Gennes energy functional in nematic liquid crystals",
  "authors": [
    "Zhewen Feng",
    "Min-Chun Hong"
  ],
  "comment": "Revised vision. To appear in Calculus of Variations and Partial Differential Equations. arXiv admin note: text overlap with arXiv:2007.11144",
  "categories": [
    "math.AP"
  ],
  "abstract": "The Landau-de Gennes energy in nematic liquid crystals depends on four elastic constants $L_1$, $L_2$, $L_3$, $L_4$. In the case of $L_4\\neq 0$, Ball and Majumdar (Mol. Cryst. Liq. Cryst., 2010) found an example that the original Landau-de Gennes energy functional in physics does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. At first, we introduce a new Landau-de Gennes energy density with $L_4\\neq 0$, which is equivalent to the original Landau-de Gennes density for uniaxial tensors and satisfies the coercivity condition for all $Q$-tensors. Secondly, we prove that solutions of the Landau-de Gennes system can approach a solution of the $Q$-tensor Oseen-Frank system without using energy minimizers. Thirdly, we develop a new approach to generalize the Nguyen and Zarnescu (Calc. Var. PDEs, 2013) convergence result to the case of non-zero elastic constants $L_2$, $L_3$, $L_4$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-09-28T23:26:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35Q35",
      "76A15"
    ],
    "keywords": [
      "nematic liquid crystals",
      "critical points",
      "convergence",
      "original landau-de gennes energy functional",
      "landau-de gennes energy density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}