On the Error of Averaging Symmetric Elliptic Systems

Abstract

Exploiting the intimate connection of the problem of a charged spin-1 particle in a homogeneous magnetic field to that of a harmonic oscillator, we demonstrate explicitly that the eigenvalues of the matrix operator S . n for spin-1 and for a constant magnetic field are governed by a Hermitian matrix defined on the space of the particle number n and spin and are thus constrained to be real for any intensity of the external magnetic field H, thereby contradicting a recent affirmation of Weaver that for n = 0, complex eigenvalues are present for sufficiently intense magnetic fields. We also point out the errors that have crept into Weaver's analysis which have led to complex eigenvalues and as well to an excessive number of eigenvalues compared with that demanded by the spin degrees of freedom.

Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems

“…Another approach to homogenization involves the use of the so-called approximate correctors [25,23]. Under certain mixing conditions, the approach has been employed successfully to establish quantitative homogenization results for second-order linear elliptic equations and systems in divergence form with random coefficients in [28,26,8]. For nonlinear second-order elliptic equations and Hamilton-Jacobi equations, we refer the reader to [9,1,2] for recent advances and references on quantitative homogenization results.…”

“…By rescaling we may assume that T = 1. The proof of the existence and estimate (5.3) may be found in [26]. It uses the fact that for f ∈ L 2 (R d ; R m ), g = (g 1 , .…”

Corrector Estimates for Elliptic Systems with Random Periodic Coefficients

“…Here, we identified Y d with the space of matrices R d×d , and by sym we denoted the symmetric part of a matrix. Finally, Pozhidaev and Yurinskiȋ [27] gave a condition which generalizes both of these conditions: they assume that for each…”

“…Quantifying ergodicity in the form of uniform mixing condition (i.e., assuming algebraic decay of correlations), Yurinskiȋ [29] was the first to prove the rate of convergence (though not optimal) of a solution to an elliptic equation with random coefficients to the solution of a homogenized equation. Later, together with Pozhidaev, Yurinskiȋ extended this result to systems of equations [27]. Assuming small ellipticity contrast ratio (requirement for the Meyers estimate to hold for exponents p = 4), in the case of a discrete elliptic equation with diagonal coefficients, Naddaf and Spencer showed in their unpublished work [25] the optimal rate of convergence.…”

Estimation of residuals for the homogenized solution of quasi-periodic media

“…Under adequate condition [5,6,7], it is known that, as the parameter scale ε → 0, the operator converge to an averaged operator with non random coefficients; this type of convergence is known as G-convergence. The accuracy of the approximation has been studied [8,9,10] but it is very difficult to adapt the proposed methods in case of special material as masonry which is characterized by a quasi-periodic texture. In order to investigate this problem, the authors, in previous papers [11,12], dealt with the case of the beam with Young's modulus randomly varying along the axis.…”