The radial-hedgehog solution in Landau-de Gennes' theory

We study the radial-hedgehog solution on a unit ball in three dimensions, with homeotropic boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a globally stable configuration in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. We use a combination of Ginzburg-Landau techniques, perturbation methods and stability analyses to study the qualitative properties of the radial-hedgehog solution, the structure of its defect core, its stability and instability with respect to biaxial perturbations. Our results complement previous work in the field, are rigorous in nature, give information about the role of geometry, elastic constants and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities

The Landau-de Gennes theory of nematic liquid crystals:\ud Uniaxiality versus Biaxiality

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 threedimensional domains separately and study the correspondence between Landau-de Gennes theory and Ginzburg-Landau theory for superconductors. We treat uniaxial and biaxial cases separately. In the uniaxial case, topological defects correspond to the zero set and we obtain results for the location and dimensionality of the defect set, the solution profile near and away from the defect set. In the three-dimensional case, we establish the C^1,a-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

Order parameters in the Landau-de Gennes theory - the static and dynamic scenarios

We obtain quantitative estimates for the scalar order parameters of liquid crystal configurations in three-dimensional geometries, within the Landau-de Gennes framework. We consider both static equilibria and non-equilibrium dynamics and we include external fields and surface anchoring energies into our formulation. Using maximum principle-type arguments, we obtain explicit bounds for the corresponding scalar order parameters in both static and dynamic situations; these bounds are given in terms of the material-dependent thermotropic coefficients, electric field strength and surface anchoring coefficients. These bounds provide estimates for the degree of orientational ordering, quantify the competing effects of the different energetic contributions and can be used to test the accuracy of numerical simulations

Inflationary universe in Kaluza-Klein theories

We describe extended inflation and its typical problems. We then briefly review essential features of Kaluza-Klein theory, and show that it leads to a scenario of inflationary cosmology in four dimensions. The problem of stable compactification of extra spatial dimensions is discussed. The requirements for successful extended inflation lead to constraints on the parameters of higher dimensional models.Comment: Latex, 13 pages (Review talk at the YATI meeting at SNBNCBS, Calcutta. To appear in Ind. Jour. Phys.

Generalized Hawking-Page Phase Transition

The issue of radiant spherical black holes being in stable thermal equilibrium with their radiation bath is reconsidered. Using a simple equilibrium statistical mechanical analysis incorporating Gaussian thermal fluctuations in a canonical ensemble of isolated horizons, the heat capacity is shown to diverge at a critical value of the classical mass of the isolated horizon, given (in Planckian units) by the {\it microcanonical} entropy calculated using Loop Quantum Gravity. The analysis reproduces the Hawking-Page phase transition discerned for anti-de Sitter black holes and generalizes it in the sense that nowhere is any classical metric made use of.Comment: 9 Pages, Latex with 2 eps figure

Symmetry of uniaxial global Landau-de Gennes minimizers in the\ud theory of nematic liquid crystals

We extend the recent radial symmetry results by Pisante [23] and Millot & Pisante [19] (who show that all entire solutions of the vector-valued Ginzburg-Landau equations in superconductivity theory, in the three-dimensional space, are comprised of the well-known class of equivariant solutions) to the Landau-de Gennes framework in the theory of nematic liquid crystals. In the low temperature limit, we obtain a characterization of global Landau-de Gennes minimizers, in the restricted class of uniaxial tensors, in terms of the well-known radial-hedgehog solution. We use this characterization to prove that global Landau-de Gennes minimizers cannot be purely uniaxial for sufficiently low temperatures

Stability estimates for a twisted rod under terminal loads: a three-dimensional study

The stability of an inextensible unshearable elastic rod with quadratic strain energy density subject to end loads is considered. A self-contained proof in terms of local energy minimizers is presented and optimal bounds are obtained for the problem