Chen, Jiajie; Zhang, Pingwen; Zhang, Zhifei Local minimizer and de Giorgi’s type conjecture for the isotropic-nematic interface problem. (English) Zbl 1400.82287 Calc. Var. Partial Differ. Equ. 57, No. 5, Paper No. 129, 19 p. (2018). Summary: In this paper, we investigate the structure of local minimizers for the isotropic-nematic interface based on the Landau-de Gennes energy. In the absence of the anisotropic energy, the uniaxial solution is the only local minimizer in 1-D. In 3-D, we propose a De Giorgi’s type conjecture and give an affirmative answer under a mild assumption. In the presence of the anisotropic energy with \(L_2>-1\) and homeotropic anchoring, the uniaxial solution is also the only local minimizer in a class of diagonal form in 1-D. Cited in 1 Document MSC: 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) 35J47 Second-order elliptic systems 35J61 Semilinear elliptic equations 82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics 82B27 Critical phenomena in equilibrium statistical mechanics Keywords:liquid crystals; isotropic-nematic interface; Landau-de Gennes energy PDFBibTeX XMLCite \textit{J. Chen} et al., Calc. Var. Partial Differ. Equ. 57, No. 5, Paper No. 129, 19 p. (2018; Zbl 1400.82287) Full Text: DOI References: [1] Ambrosio, L.; Cabré, X., Entire solutions of semilinear elliptic equations in \({\mathbf{R}}^3\) and a conjecture of De Giorgi, J. Am. Math. Soc., 13, 725-739, (2000) · Zbl 0968.35041 · doi:10.1090/S0894-0347-00-00345-3 [2] Ball, JM; Majumdar, A., Nematic liquid crystals: from maier-saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525, 1-11, (2010) · doi:10.1080/15421401003795555 [3] de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1993) [4] Pino, M.; Kowalczyk, M.; Wei, J., On De Giorgi’s conjecture in dimension \(N\ge 9\), Ann. Math., 174, 1485-1569, (2011) · Zbl 1238.35019 · doi:10.4007/annals.2011.174.3.3 [5] Fazly, M.; Ghoussoub, N., De Giorgi type results for elliptic systems, Calc. Var. Partial Differ. Equ., 47, 809-823, (2013) · Zbl 1275.35094 · doi:10.1007/s00526-012-0536-x [6] Faetti, S.; Palleschi, V., Molecular orientation and anchoring energy at the nematic-isotropic interface of 7CB, J. Phys. Lett., 45, 313, (1984) · doi:10.1051/jphyslet:01984004507031300 [7] Ghoussoub, N.; Gui, C., On a conjecture of De Giorgi and some related problems, Math. Ann., 311, 481-491, (1998) · Zbl 0918.35046 · doi:10.1007/s002080050196 [8] Kamil, SM; Bhattacharjee, AK; Adhikari, R.; Menon, GI, Biaxiality at the isotropic-nematic interface with planar anchoring, Phys. Rev. E, 80, 041705, (2009) · doi:10.1103/PhysRevE.80.041705 [9] Kamil, SM; Bhattacharjee, AK; Adhikari, R.; Menon, GI, The isotropic-nematic interface with an oblique anchoring condition, J. Chem. Phys., 131, 174701, (2009) · doi:10.1063/1.3253702 [10] Park, J.; Wang, W.; Zhang, P.; Zhang, Z., On minimizers for the isotropic-nematic interface problem, Calc. Var. Partial Differ. Equ., 56, 41, (2017) · Zbl 1366.82060 · doi:10.1007/s00526-017-1131-y [11] Popa-Nita, V.; Sluckin, TJ; Wheeler, AA, Statics and kinetics at the nematic-isotropic interface: effects of biaxiality, J. Phys. II (France), 7, 1225-1243, (1997) · doi:10.1051/jp2:1997183 [12] Savin, O., Regularity of flat level sets in phase transitions, Ann. Math., 169, 41-78, (2009) · Zbl 1180.35499 · doi:10.4007/annals.2009.169.41 [13] Sternberg, P., Vector-valued local minimizers of nonconvex variational problems, Rocky Mt. J. Math., 21, 799-807, (1991) · Zbl 0760.49008 · doi:10.1216/rmjm/1181072968 [14] Wang, W.; Zhang, P.; Zhang, Z., Rigorous derivation from Landau-de Gennes theorey to Ericksen-Leslie theory, SIAM J. Math. Anal., 47, 127-158, (2015) · Zbl 1335.35205 · doi:10.1137/13093529X This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.