Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals

We analyze the radially symmetric solution corresponding to the vortex defect (the so-called melting hedgehog) in the Landau–de Gennes theory for nematic liquid crystals. We prove the existence, uniqueness and stability results of the melting hedgehog.     

Asymptotic stability and instability of the solutions of systems with impulse action

In this paper, we study the stability of the zero solution of a system of ordinary differential equations subject to impulse action. Using the method of Lyapunov functions, we obtain tests for asymptotic stability or instability of the system. Illustrative examples are given.     

Approximation of Analytic Functions with an Arbitrary Order by Generalized Baskakov–Faber Operators in Compact Sets

By using a sequence \(\lambda _{n}>0\), \(n\in \mathbb {N}\) with the property that \(\lambda _{n}\rightarrow 0\) as fast we want, in this paper we obtain the approximation order \(O(\lambda _{n})\) for a generalized Baskakov–Faber operator attached to analytic functions of exponential growth in a continuum \(G\subset \mathbb {C}\). Several concrete examples of continuums G are given for which this operator can explicitly be constructed.     

On the impact of an event

Given an information structure a function that measures how the inferences made by the agents spread among the states is defined; it specifies for each event its impact at each state, for each agent. Several properties are established. Received November 1994/Revised version September 1996     

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R²→S² that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S¹-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}_ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-vector fields by S¹-vector fields away from the vortex balls.     

A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation which are periodic in the d − 1 last variables (living on the torus 𝕋^d−1) and globally minimize the corresponding energy in Ω = ℝ × 𝕋^d−1, i.e., Namely, we find a class of nonlinear potentials W ≥ 0 such that any global minimizer u of E connecting two zeros of W as x₁ → ± ∞ is one-dimensional; i.e., u depends only on the x₁ -variable. In particular, this class includes in dimension d = 2 the nonlinearities with w being a harmonic function or a solution to the wave equation, while in dimension d ≥ 3 , this class contains a perturbation of the Ginzburg-Landau potential as well as potentials W having d + 1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of E for general potentials W providing in particular compactness results for uniformly finite energy maps u in Ω connecting two wells of W as x₁ → ± ∞ . © 2019 Wiley Periodicals, Inc.