1,720,993 research outputs found
An estimate of the blow-up of Lebesgue norms in the non-tempered case
No full textWe prove that if p>1 and ψ:]0,p−1]→]0,∞[ is just nondecreasing and differentiable (hence not necessarily Δ2), then for every f Lebesgue measurable function on (0,1) sup0<1Sψ(t)‖f⁎‖Ljavax.xml.bind.JAXBElement@5695bc9b(t,1), where f⁎ denotes the decreasing rearrangement of f and Sψ is defined, for ε∈]0,p−1[, through [Formula presented] where cψ is the normalizing constant chosen so that ν((p−1)−)=1. If ψ is in a class of functions satisfying the Δ2 condition, essentially characterized by the so-called ∇′ condition, then inequality (⁎) is sharp, in the sense that both sides are equivalent. Estimate (⁎) generalizes an inequality of the type obtained by the second author with Farroni and Giova in [6] under the growth condition ψ∈Δ2
On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces
No full textThe paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of S2-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are z-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaŕe-type inequality on the circular cylinder, which allows for establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain
On a sharp Poincare-type inequality on the 2-sphere and its application in micromagnetics
No full textThe main aim of this note is to prove a sharp Poincare-type inequality for vector-valued functions on S2 that naturally emerges in the context of micromagnetics of spherical thin films
SYMMETRY PROPERTIES OF MINIMIZERS OF A PERTURBED DIRICHLET ENERGY WITH A BOUNDARY PENALIZATION
No full textWe considerS 2-valued maps on a domain Ω C R N minimizing a perturbation of the Dirichlet energy with vertical penalization in Ω and horizontal penalization on Ω . We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincar\'e-type inequality. Then we prove that any energy minimizer takes its values into a fixed half-meridian of the sphere S 2 and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minimizers with different penalizations. Finally, we apply these results to a problem on a ball and show radial symmetry and monotonicity of minimizers. In dimension N = 2 our results can be applied to the Oseen-Frank energy for nematic liquid crystals and the micromagnetic energy in a thin-film regime
Reduced Models for Ferromagnetic Thin Films with Periodic Surface Roughness
Full text linkWe investigate the influence of periodic surface roughness in thin ferromagnetic films on shape anisotropy and magnetization behavior inside the ferromagnet. Starting from the full micromagnetic energy and using methods of homogenization and (Formula presented.)-convergence, we derive a two-dimensional local reduced model. Investigation of this model provides an insight into the formation mechanism of perpendicular magnetic anisotropy and uniaxial anisotropy with an arbitrary preferred direction of magnetization.</p
Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals
No full textWe study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of Γ -development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing Q-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy
Geometrically induced phase transitions in two-dimensional dumbbell-shaped domains
Full text linkWe continue the analysis, started in [23], of a two-dimensional non-convex variational problem, motivated by studies on magnetic domain walls trapped by thin necks. The main focus is on the impact of extreme geometry on the structure of local minimizers representing the transition between two different constant phases. We address here the case of general non-symmetric dumbbell-shaped domains with a small constriction and general multi-well potentials. Our main results concern the existence and uniqueness of non-constant local minimizers, their full classification in the case of convex bulks, and the complete description of their asymptotic behavior, as the size of the constriction tends to zero.</p
Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals
No full textWe investigate prototypical profiles of point defects in two-dimensional liquid crystals within the framework of Landau–de Gennes theory. Using boundary conditions characteristic of defects of index k/2, we find a critical point of the Landau–de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime, b2 small, we prove that this critical point is the unique global minimiser of the Landau–de Gennes energy. For the case b2=0, we investigate in greater detail the regime of vanishing elastic constant L→0, where we obtain three explicit point defect profiles, including the global minimiser
Engineering Curvature-Induced Anisotropy in Thin Ferromagnetic Films
No full textWe investigate the effect of large curvature and dipolar energy in thin ferromagnetic films with periodically modulated top and bottom surfaces on magnetization behavior. We predict that the dipolar interaction and surface curvature can produce perpendicular anisotropy which can be controlled by engineering special types of periodic surface structures. Similar effects can be achieved by a significant surface roughness in the film. We demonstrate that, in general, the anisotropy can point in an arbitrary direction depending on the surface curvature. Furthermore, we provide simple examples of these periodic surface structures to show how to engineer particular anisotropies in thin films
Spin-diffusion model for micromagnetics in the limit of long times
No full textIn this paper, we consider spin-diffusion Landau–Lifshitz–Gilbert equations (SDLLG), which consist of the time-dependent Landau–Lifshitz–Gilbert (LLG) equation coupled with a time-dependent diffusion equation for the electron spin accumulation. The model takes into account the diffusion process of the spin accumulation in the magnetization dynamics of ferromagnetic multilayers. We prove that in the limit of long times, the system reduces to simpler equations in which the LLG equation is coupled to a nonlinear and nonlocal steady-state equation, referred to as SLLG. As a by-product, the existence of global weak solutions to the SLLG equation is obtained. Moreover, we prove weak-strong uniqueness of solutions of SLLG, i.e., all weak solutions coincide with the (unique) strong solution as long as the latter exists in time. The results provide a solid mathematical ground to the qualitative behavior originally predicted by ZHANG, LEVY, and FERT in [44] in ferromagnetic multilayers
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