Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

Abstract

The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.

A Lyapunov type inequality for indefinite weights and eigenvalue homogenization

“…We may ask what happens when ρ is an indefinite weight, and our next Theorem generalizes to the one dimensional quasilinear setting the answer for second order linear problems (in arbitrary spatial dimension) obtained recently by Nazarov, Pankratova and Piatnitski in [16,18]:…”

“…Let us remark that in the first case, after a suitable renormalization µ ± ε,k = ε α λ ± ε,k , the convergence to the eigenvalues of a different limit problem was obtained in [16,18]. Their proofs were based on linear tools such as orthogonality of eigenfunctions or asymptotic expansions in powers of ε which are not available here.…”

Homogenization of the Spectral Equation in One Dimension

“…Theorem 9 For k ∈ Y * , let (λ ε , w ε ) be solution of ( 8) then (23,24), assuming that the weak limit of S ε k w ε in L 2 (Ω; H 1 (Y )) is non-vanishing and the renormalized sequence ε 2 λ ε satisfies the decomposition (10), there exists n ∈ N * such that λ 0 = λ k n with λ k n an eigenvalue of the Bloch wave spectrum and the limit g k of any weakly converging extracted subsequence of…”

“…The low frequency part of the spectrum has been investigated in [17], [18], [25]. Then, many configurations have been analyzed, as [16] and [13] for a fluid-structure interaction, [7], [3] for neutron transport, [22], [24] for ρ which changes sign or [4] for the first high frequency eigenvalue and eigenvector for a one-dimensional non-self-adjoint problem with Neumann boundary conditions. In [6], G. Allaire and C. Conca studied the asymptotic behaviour of both the low and high frequency spectrum.…”

Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

“…Boundary value and spectral problems in thin domains are usually treated using the analysis of resolvents ( [FS09]), the method of asymptotic expansions (see for example [CD79], [Pan05], [BF10], [MP10], [Naz01], [PS13]), two-scale convergence ( [EP96], [MMP00], [PP11], [PP15]), Γ-convergence ( [MS95], [AB01], [BFF00], [Gau+02], [BMT07], [BMT12]), compensated compactness agrument ( [GM03]), and the unfolding method ( [BG08], [AP11], [AVP17]). The presented list of works devoted to the homogenization in thin structures is far from being complete, but our primary focus is the case of thin domains with locally periodic rapidly varying thickness, and to our best knowledge the works closely related to our study are [MP10], [AP11], [FS09], [BF10], and [NPT16].…”