We study the Schrodinger operator (hD-A)(2) with periodic magnetic field B=curl A in an antidot lattice Omega(infinity) = R-2\boolean OR(alpha is an element of Gamma)(U+alpha). Neumann boundary conditions lead to spectrum below hinf B. Under suitable assumptions on a "one-well problem" we prove that this spectrum is localized inside an exponentially small interval in the semi-classical limit h -> 0. For this purpose we construct a basis of the corresponding spectral subspace with natural localization and symmetry properties.

We consider energy functionals, or Dirichlet forms, [GRAPHICS] for a class G of bounded domains Omega subset of R-N, with epsilon>0 a fine structure parameter and with symmetric conductivity matrices A(epsilon) = (a(ij)(epsilon)) is an element of L-loc(infinity)(R)(NxN) which are functions only of the first coordinate x(1) and which are locally uniformly elliptic for each fixed epsilon>0. We show that if the functions (of x(1)) b(11)(epsilon) = 1/a(11)(epsilon), b(1j)(epsilon) = a(1j)(epsilon)/a(11)(epsilon) (j greater than or equal to 2), b(ij)(epsilon) = a(ij)(epsilon) - a(i1)(epsilon)a(1j)(epsilon)/ a(11)(epsilon) (i, j greater than or equal to 2) converge weakly* as measures towards corresponding limit measures b(ij) as epsilon --> 0, if the (1,1)-coefficient m(11)(epsilon) of (A(epsilon))(-1) is bounded in L-loc(1)(R) and if none of its weak* cluster measures has atoms in common with b(ii), i greater than or equal to 2, then the family J(epsilon) = {J(Omega)(epsilon)}(Omega is an element of g) Gamma-converges in a local sense towards a naturally defined limit family J = {J(Omega))(Omega is an element of G) as epsilon-->0. An alternative way of formulating the conclusion is to say that the energy densities (A(epsilon)del u,del u) Gamma-converge in a distributional sense towards the corresponding limit density. Writing J(Omega)(epsilon) in terms of B-epsilon = (b(ij)(epsilon)) it becomes [GRAPHICS] and the definition of J(Omega) and the limit density (A del u, del u) is obtained by properly replacing the b(ij)(epsilon) is an element of L-loc(infinity)(R) by the limit measures b(ij) and making sense to everything for u in a certain linear subspace of L-loc(2)(R-N).

3.

Gustafsson, Björn

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

The aim of this paper is to extend, to the linear elasticity system, the asymptotic analysis by compensated compactness previously developed by the authors for the linear diffusion equation. For simplicity, we restrict ourselves to stratified media. In the case of sole homogenization we recover the classical result of W.H. Mc Connel, deriving explicitly the effective elasticity tensor for stratified media. Here we give a new proof of his result, based on compensated compactness and on a technique of decomposing matrices. As for the case of simultaneous homogenization and reduction of dimension, we perform the asymptotic analysis, as the thickness tends to zero, of a three-dimensional laminated thin plate having an anisotropic, rapidly oscillating elasticity tensor. The limit problem is presented in three different ways, the final formulation being a fourth-order problem on the two-dimensional plate, with explicitly given elasticity tensors and effective source terms.

4.

Gustafsson, Björn

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We give a criterion for H-convergence of elasticity tensors in terms of ordinary weak convergence of the factors in certain quotient representations of the tensors.

5.

Laptev, Ari

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Safronov, Oleg

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We obtain an asymptotic formula for the number of negative eigenvalues of a class of two-dimensional Schrodinger operators with small magnetic fields. This number increases as a coupling constant of the magnetic field tends to zero.