Using Landau-de Gennes theory to describe nematic order, we study a frustrated cell consisting of nematic liquid crystal confined between two parallel plates. We prove the uniqueness of equilibrium states for a small cell width. Letting the cell width grow, we study the behaviour of this unique solution. Restricting ourselves to a certain interval of temperature, we prove that this solution becomes unstable at a critical value of the cell width. Moreover, we show that this loss of stability comes with the appearance of two new solutions: there is a symmetric pitchfork bifurcation. This picture agrees with numerical simulations performed by P. Palffy-Muhorray, E.C. Gartland and J.R. Kelly. Some of the methods that we use in the present paper apply to other situations, and we present the proofs in a general setting. More precisely, the paper contains the proof of a general uniqueness result for a class of perturbed quasilinear elliptic systems, and general considerations about symmetric solutions and their stability, in the spirit of Palais' Principle of Symmetric Criticality.

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In the present work, we study minimizers of the Landau-de Gennes free energy in a bounded domain Ω ⊂ R 3 . We prove that at low temperature minimizers do not vanish, even for topologically non-trivial boundary conditions. This is in contrast with a simplified Ginzburg-Landau model for superconductivity studied by Bethuel, Brezis and Hélein. Merging this with an observation of Canevari we obtain, as a corollary, the occurence of biaxial escape: the tensorial order parameter must become strongly biaxial at some point in Ω. In particular, while it is known that minimizers cannot be purely uniaxial, we prove the much stronger and physically relevant fact that they lie in a different homotopy class.

We consider a nematic liquid crystal occupying the exterior region in R 3 outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to a strong external field, modelled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.

In this paper we study the eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production, and the validity of a kinetic formulation. © 2019 Wiley Periodicals, Inc.

a b s t r a c tWe consider, in the Landau-de Gennes theoretical framework of a Q -tensor description of nematic liquid crystals, a radial hedgehog defect with strong anchoring conditions in a ball B ⊂ R 3 . We show that the scalar order parameter is monotonic, and we prove uniqueness of the minimizing hedgehog below the spinodal temperature T * .

In co-manipulation, humans and robots solve manipulation tasks together. Virtual guides are important tools for co-manipulation, as they constrain the movement of the robot to avoid undesirable effects, such as collisions with the environment. Defining virtual guides is often a laborious task requiring expert knowledge. This restricts the usefulness of virtual guides in environments where new tasks may need to be solved, or where multiple tasks need to be solved sequentially, but in an unknown order. To this end, we propose a framework for multiple probabilistic virtual guides, and demonstrate a concrete implementation of such guides using kinesthetic teaching and Gaussian mixture models. Our approach enables non-expert users to design virtual guides through demonstration. Also, they may demonstrate novel guides, even if already known guides are active. Finally, users are able to intuitively select the appropriate guide from a set of guides through physical interaction with the robot. We evaluate our approach in a pick-and-place task, where users are to place objects at one of several positions in a cupboard.

Virtual guiding fixtures constrain the movements of a robot to task-relevant trajectories, and have been successfully applied to, for instance, surgical and manufacturing tasks. Whereas previous work has considered guiding fixtures for single tasks, in this paper we propose a library of guiding fixtures for multiple tasks, and propose methods for 1) Creating and adding guides based on machine learning; 2) Selecting guides on-line based on probabilistic implementation of guiding fixtures; 3) Refining existing guides based on an incremental learning method. We demonstrate in an industrial task that a library of guiding fixtures provides an intuitive haptic interface for joint human-robot completion of tasks, and improves performance in terms of task execution time, mental workload and errors.