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Abstract We construct an entire solution to the elliptic systemwhere is a ‘triple‐well’ potential. This solution is a local minimizer of the associated energyin the sense that minimizes the energy on any compact set among competitors agreeing with outside that set. Furthermore, we show that along subsequences, the ‘blowdowns’ of given by approach a minimal triple junction as . Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions. 

It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg–Landau equations-\Delta u_{\varepsilon} +\varepsilon^{-2}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon} = 0, the energy and vorticity concentrate as\varepsilon\to 0around a codimension2stationary varifold – a (measure-theoretic) minimal surface. Much less is known about the question of whether, given a codimension2minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen–Cahn equation, and for the Ginzburg–Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open. We consider this question on a3-dimensional closed Riemannian manifold(M,g), and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/ vorticity concentration set of a sequence of solutions of the Ginzburg–Landau equations. 

The domain structure of a fluid ferroelectric nematic is dramatically different from the domain structure of solid ferroelectrics since it is not restricted by rectilinear crystallographic axes and planar surface facets. We demonstrate that thin films of a ferroelectric nematic seeded by colloidal inclusions produce domain walls (DWs) in the shape of conics such as a parabola. These conics reduce the bound charge within the domains and at the DWs. An adequate description of the domain structures requires one to analyze the electrostatic energy, which is a challenging task. Instead, we demonstrate that a good approximation to the experimentally observed polydomain textures is obtained when the divergence of spontaneous polarization—which causes the bound charge—is heavily penalized by assuming that the elastic constant of splay in the Oseen-Frank energy is much larger than those for twist and bend. The model takes advantage of the fact that the polarization vector is essentially parallel to the nematic director throughout the sample. Published by the American Physical Society2024