In this talk, we study a ferromagnetic nanowire with a defect, represented as a single, unimodal notch. Using a mountain-path argument, we establish the existence and uniqueness of a critical point for the ferromagnetic energy associated with this model. This critical point corresponds to a topological solution (a single domain wall) localized in the vicinity of the notch. This work allows the...
Korn's inequality and its variants are essential tools in the mathematical analysis of both linear and nonlinear elasticity. They play a central role in establishing existence and regularity results for partial differential equations involving symmetric gradients. In this talk, I will present a conceptually simple derivation of the first and second Korn inequalities for general exponents $1 <...
This talk is the first of two parts, presented jointly with Federico Luigi Dipasquale. Ferronematics are composite materials characterised by the coupling between magnetic particles and nematic liquid crystals. In these talks, we will present some results on a two-dimensional model for ferronematics in confined geometries. The model is based on the coupling between a polar order parameter –...
We consider a variational model for ferronematics --- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes $Q$-tensor for the liquid crystal component and a magnetisation vector field $M$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling...
The Ericksen model describes nematic liquid crystals (LCs) in terms of a unit-length vector field (director) and a scalar function (degree of orientation). Compared to the classical Oseen-Frank model, it allows for the description of a larger class of defects. Equilibrium states of the LC are given by admissible pairs that minimize an energy functional, which consists of the sum of...
