We study equilibrium configurations of nematic liquid crystals confined to two-dimensional isosceles triangles, subject to tangent boundary conditions. This toy problem is motivated by the effects of geometrical asymmetry on equilibria in variational problems arising in liquid crystal theory. There are two key geometrical parameters for an isosceles triangle - the triangle edge length and the apex angle. The nematic equilibria are modelled by minimizers of a reduced Landau-de Gennes free energy in this setting. For small edge lengths, we provide a universal, angle-based local classification of nematic equilibria near the vertices as to whether the nematic director exhibits a splay, bend or singular profile depending on the vertex opening angle. In the large domain limit, we demonstrate the existence of multiple competing nematic equilibria — the three rotated solutions, for which the nematic director bends between a pair of adjacent vertices, and a trefoil solution featuring an interior point defect. For acute apex angles, we show that the trefoil solution is stable for small edge lengths. The interior point defect of the trefoil solution migrates to one of the base vertices, as the edge length increases, and is finally expelled giving way to the rotated solutions, if the apex angle is small enough. Our numerical results suggest that there is a unique trefoil solution on the equilateral triangle for all edge lengths, and a unique rotated solution on isosceles triangles with wide apex angles. These results yield interesting insight into how geometrical asymmetry can tailor equilibria and self-assembly processes in confined nematic systems.

We present a rigorous mathematical analysis of a modified Landau-de Gennes (LdG) theory modeling temperature-driven phase transitions between cholesteric, helical smectic, and smectic C* phases. This model couples a tensor-valued order parameter (nematic orientational order) with a real-valued order parameter (smectic layer modulation). We establish the existence of energy minimizers of the modified LdG energy in three dimensions, subject to Dirichlet conditions, and rigorously analyze the energy minimizers in two asymptotic limits. First, in the Oseen--Frank limit, we show that the global minimizer strongly converges to a minimizer of the Landau-de Gennes bulk energy. Second, in the limit of dominant elastic constants, we prove that the global minimizers converge to a classical helical director profile. Finally, through stability analysis and bifurcation theory, we derive the complete sequence of symmetry-breaking transitions with decreasing temperature-from the cholesteric phase (with in-plane twist and no layering) to an intermediate helical smectic phase (with in-plane twist and layering), and ultimately to the smectic C* phase (with out-of-plane twist and layering). These theoretical results are supported by numerical simulations.

We study the hydrodynamics of compressible active nematic liquid crystals in a three-dimensional and bounded domain, with a nonlinear viscosity tensor and nonhomogeneous boundary data, in a Landau-de Gennes framework. We prove the existence of dissipative solutions within a Beris-Edwards type model for active nematodynamics, which are weak solutions satisfying the underlying equations modulo a defect measure. The proof follows from a three level approximation scheme -- the Galerkin approximation, the classical parabolic regularization of the continuity equation, and the convex regularization of the potential generating the viscous stress. New techniques are required to deal with non-Newtonian stress tensor, larger classes of admissible pressure potentials and nonhomogeneous boundary conditions.