We address the effect of extreme geometry on a non-convex variational problem, motivated by studies on magnetic domain walls trapped by thin necks. The recent analytical results of Kohn and Slastikov revealed a variety of magnetic structures in three-dimensional ferromagnets depending on the size of the constriction. The main purpose of the paper is to study geometrically constrained walls in two dimensions. The analysis turns out to be significantly more challenging and requires the use of different techniques. In particular, the purely variational point of view cannot be adopted in the present setting and is here replaced by a PDE approach. Existence of local minimizers representing geometrically constrained walls is proven under suitable symmetry assumptions on the domains and an asymptotic characterization of the wall profile is given. The limiting behavior, which depends critically on the scaling of length and height of the neck, turns out to be more complex than in the higher-dimensional case and a richer variety of regimes is shown to exist.

We show that recently reported precessing solution of Landau-Lifshitz-Gilbert equations in ferromagnetic nanowires is stable under small perturbations of initial data, applied field and anisotropy constant. Linear stability is established analytically, while nonlinear stability is verified numerically.

In this paper, we consider the micromagnetic variational problem for soft ferromagnetic nanowires. We show that, as the diameter of the wire is small, the magnetization inside the wire depends only on the length variable of the wire. The micromagnetic energy of the wire, in this case, is greatly simplified and in order to find the optimal magnetization distribution, one has to solve a 1D local variational problem.

We address the dynamics of magnetic domain walls in ferromagnetic nanowires under the influence of external time-dependent magnetic fields. We report a new exact spatiotemporal solution of the Landau-Lifshitz-Gilbert equation for the case of soft ferromagnetic wires and nanostructures with uniaxial anisotropy. The solution holds for applied fields with arbitrary strength and time dependence. We further extend this solution to applied fields slowly varying in space and to multiple domain walls.

We investigate an effect of configurational anisotropy in highly symmetric soft ferromagnetic nanoparticles. Using micromagnetic variational principle and methods of Gamma-convergence we show that in ferromagnetic prisms with D4 symmetry there is a finite number of preferred magnetization directions and that these directions are independent of the shape of a magnet. This result provides a rigorous justification of work by Cowburn and Welland.

We address the effect of extreme geometry on a non-convex variational problem. The analysis is motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. To capture the essential issues in the simplest possible setting, we focus on a scalar variational problem with a symmetric double well potential, whose spatial domain is a dumbell with a sharp neck. Our main results are (a) the existence of local minimizers representing geometrically constrained walls, and (b) an asymptotic characterization of the wall profile. Our analysis uses methods similar to Gamma-convergence; in particular, the wall profile minimizes a certain ``reduced problem'' -- the limit of the original problem, suitably rescaled near the neck. The structure of the wall depends critically on the choice of scaling, specifically the ratio between length and width of the neck.

We study a thin-shell limit of micromagnetic energy for soft small ferromagnets. The relations between thickness of the magnet t , diameter l and magnetic exchange length w are t/l --> 0 and tl/w^2 < 1. We prove a Gamma-convergence of the original 3D problem to a nonlocal 2D problem.

We consider the variational problem of micromagnetics for soft, relatively small thin films with no applied magnetic field. In terms of the film thickness t , the diameter l and the magnetic exchange length w, we study the asymptotic behavior in the small-aspect-ratio limit t/l --> 0, when w^2/l^2 >= (t/l) | log (t/l) | . Our analysis builds on prior work by Gioia & James and Carbou. The limiting variational problem is much simpler than 3D micromagnetics; in particular it is two-dimensional and local, with no small parameters.

In a thin-film ferromagnet, the leading-order behavior of the magnetostatic energy is a strong shape anistropy, penalizing the out-of-plane component of the magnetization distribution. We study the thin-film limit of Landau-Lifshitz-Gilbert dynamics, when the magnetostatic term is replaced by this local approximation. The limiting 2D effective equation is overdamped, i.e. it has no precession term. Moreover if the damping coefficient of 3D micromagnetics is a then the damping coefficient of the 2D effective equation is a + 1/a; thus reducing the damping in 3D can actually increase the damping of the effective equation. This result was previously shown by Garcia-Cervera and E using asymptotic analysis; our contribution is a mathematically rigorous justification.

We study a two-dimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended Onsager- Maier-Saupe free energy may be decomposed into Landau-de Gennes-type and relative entropy-type contributions. We then prove that in the high concentration limit the states of the system display characteristic vortex-like patterns and derive asymptotic expansion for the free energy of the system.

We present a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the suggested free energy is a functional of spatially-dependent orientation distribution, however, the nonlocal effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. In illustration of this approach we consider a simplified model with orientation parameter on a circle and reveal its relation to the complex GinzburgÐLandau theory.

We study OnsagerÕs model of isotropicÐnematic phase transitions with orientation parameter on a sphere. We consider two interaction potentials: the antisymmetric (with respect to orientation inversion) dipolar potential and symmetric MaierÐSaupe potential. We prove the axial symmetry and derive explicit formulae for all critical points, thus obtaining their complete classification. Finally, we investigate their stability and construct the corresponding bifurcation diagrams.

We study Onsager's free energy functional for nematic liquid crystals with an orien- tation parameter on a unit circle. For a class of interaction potentials we obtain explicit expressions for all critical points, analyze their stability, and construct the corresponding bifurcation diagram. We also derive asymptotic expansions of the equilibrium density of orientations near the critical and zero temperatures.

The role of surfactants in stabilizing the formation of bubbles in foams is studied using a phase-field model The analysis is centered on a van der Walls-Cahn-Hilliard-type energy with an added term accounting for the interplay between the presence of a surfactant density and the creation of interfaces. In particular, it is concluded that the surfactant segregates to the interfaces, and that the prescription of the distribution of surfactant will dictate the locus of interfaces, what is in agreement with experimentation.