Liouville type results for local minimizers of the micromagnetic energy

Abstract

International audienceWe study local minimizers of the micromagnetic energy in small ferromagnetic 3d convex particles for which we justify the Stoner-Wohlfarth approximation: given a uniformly convex shape $\Omega \subset {\mathbf{R}}^3$, there exist $\delta_c$>0 and $C > 0$ such that for $0 < \delta \leq \delta_c$ any \textit{local} minimizer $\mathbf{m}$ of the micromagnetic energy in the particle $\delta \Omega$ satisfies $\|\nabla \mathbf{m} \|_{L^2} \leqslant C \delta^2$.In the case of ellipsoidal particles, we strengthen this result by proving that, for $\delta$ small enough, \tmtextit{local} minimizers are exactly spatially uniform.This last result extends W.F. Brown's fundamental theorem for fine 3d ferromagnetic particles [Brown (1968), Di Fratta et al. (2011)] which states the same result but only for \textit{global} minimizers.As a by-product of the method that we use, we establish a new Liouville type result for locally minimizing $p$-harmonic maps with values into a closed subset of a Hilbert space. Namely, we establish that in a smooth uniformly convex domain of $\mathbf{R}^d$ any local minimizer of the $p$-Dirichlet energy ($p > 1$, $p \neq d$) is constant

On a Sharp Poincaré-Type Inequality on the 2-Sphere and its Application in Micromagnetics

“…We have the following definition (see [10]). (1) n,j , y (2) n,j , and y (3) n,j of degree n and order j, with (n, j) ∈ J, are defined by y (1) n,j (ξ) := Y n,j (ξ)n(ξ), y (2) n,j (ξ) :=…”

“…As the minimizers of our problem will be fully characterized in terms of the first vector spherical harmonics, it is worth to explicitly write down their explicit expressions. By the relation y (1) n,j (ξ) := Y n,j (ξ)n(ξ) we get, for n = 0, that y (1)…”

“…Hence, by making use of the relations (cf. [10, p.237]) −∆ * y (1) n,j = (n * + 2) y (1) n,j − 2 √ n * y (2) n,j ,…”

The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

“…Due to its physical relevance, particular attention is paid to the eigenvalues of P . Indeed, when N = 3, the matrix P and its eigenvalues, known in the theory of ferromagnetism respectively as the demagnetization tensor and the demagnetizing factors, are one of the most important and well-studied quantities of ferromagnetism [1,2,5,7,10,25]. In fact, the following magnetostatic counterpart of the homogeneous ellipsoid problem holds: given a uniformly magnetized ellipsoid, the induced magnetic field is also uniform inside the ellipsoid .…”

The Newtonian potential and the demagnetizing factors of the general ellipsoid

“…In §3, we focus attention to the three-dimensional case. An expression in terms of the elliptic integrals is given for the coefficients of P. Owing to its physical relevance, particular attention is paid to the eigenvalues of P. Indeed, when N = 3, the matrix P and its eigenvalues, known in the theory of ferromagnetism, respectively, as the demagnetization tensor and the demagnetizing factors, are one of the most important and well-studied quantities of ferromagnetism [20,21,23,28,29,33]. In fact, the following magnetostatic counterpart of the homogeneous ellipsoid problem holds: given a uniformly magnetized ellipsoid, the induced magnetic field is also uniform inside the ellipsoid.…”