Asymptotics of Landau-de Gennes theory

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Description:	Robbins, J Monday 14th January 2019 - 11:00 to 11:45


Created:	2019-01-15 13:44
Collection:	The mathematical design of new materials
Publisher:	Isaac Newton Institute
Copyright:	Robbins, J
Language:	eng (English)


Abstract:	We consider the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain subject to integer-degree boundary conditions, consistent with the absence of defects, in the physically relevant regime of weak elasticity. At leading order, the minimum-energy configuration is described by the simpler Oseen-Frank theory. We obtain the next-order corrections using a Gamma-convergence approach. These turn out to be determined by an algebraic rather than a differential equation. The most important qualitative feature is the appearance of biaxiality, with strength and orientation determined by the gradient of the Frank director. The results are applied to the variational problem in which only the degree of the boundary conditions is fixed. In contrast to an analogous and well-known problem in the Ginzburg-Landau model of vortices, it is found that the energy is only partially degenerate at leading order, with a family of conformal boundary conditions, parameterised by the positions of escape points (the analogues of vortices), achieving the minimum possible energy. This partial degeneracy is lifted at the next order. This is joint work with G di Fratta, V Slastikov and A Zarnescu.

Abstract:

We consider the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain subject to integer-degree boundary conditions, consistent with the absence of defects, in the physically relevant regime of weak elasticity. At leading order, the minimum-energy configuration is described by the simpler Oseen-Frank theory. We obtain the next-order corrections using a Gamma-convergence approach. These turn out to be determined by an algebraic rather than a differential equation. The most important qualitative feature is the appearance of biaxiality, with strength and orientation determined by the gradient of the Frank director. The results are applied to the variational problem in which only the degree of the boundary conditions is fixed. In contrast to an analogous and well-known problem in the Ginzburg-Landau model of vortices, it is found that the energy is only partially degenerate at leading order, with a family of conformal boundary conditions, parameterised by the positions of escape points (the analogues of vortices), achieving the minimum possible energy. This partial degeneracy is lifted at the next order. This is joint work with G di Fratta, V Slastikov and A Zarnescu.

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