Homogenization of the Spectral Problem for Periodic Elliptic Operators with Sign-Changing Density Function

Назаров, С. А.; Pankratova, Iryna; Piatnitski, Andrey

doi:10.1007/s00205-010-0370-2

Cited by 12 publications

(26 citation statements)

References 14 publications

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“…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]:…”

Section: Introductionmentioning

confidence: 67%

“…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.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Lyapunov type inequality for indefinite weights and eigenvalue homogenization

Bonder

Pinasco

Salort

2015

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

A Lyapunov type inequality for indefinite weights and eigenvalue homogenization

Bonder

Pinasco

Salort

2015

Proc. Amer. Math. Soc.

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“…However, the presence of sign-changing weight function makes the problem nonstandard and leads to new interesting phenomena. For operators with pure periodic coefficients defined in a fixed (not asymptotically thin) domains similar problems have been studied in the recent works [11,12]. In contrast with problems investigated in these works, for the model considered in the present paper the limit spectral problem is one-dimensional, so that dimension reduction arguments are to be used.…”

Section: Introductionmentioning

confidence: 94%

“…To study this operator pencil we apply the results from [8] combined with usual arguments used when studying Sturm-Liouville problems. It should be noted that in contrast with [12], the presence of slow variable in the coefficients makes the limit operator pencil nontrivial, so that it cannot be reduced to the standard Sturm-Liouville problem.…”

Section: Introductionmentioning

confidence: 98%

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

Pankratova

Piatnitski

2011

Journal of Differential Equations

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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.

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Spectrum of a diffusion operator with coefficient changing sign over a small inclusion

Chesnel

Claeys

Назаров

2015

Z. Angew. Math. Phys.

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We study a spectral problem (P δ ) for a diffusion like equation in a 3D domain Ω. The main originality lies in the presence of a parameter σ δ , whose sign changes on Ω, in the principal part of the operator we consider. More precisely, σ δ is positive on Ω except in a small inclusion of size δ > 0. Because of the sign-change of σ δ , for all δ > 0 the spectrum of (P δ ) consists of two sequences converging to ±∞. However, at the limit δ = 0, the small inclusion vanishes so that there should only remain positive spectrum for (P δ ). What happens to the negative spectrum? In this paper, we prove that the positive spectrum of (P δ ) tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of (P δ ) behaves like δ −2 µ for some constant µ < 0. We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results.

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Homogenization of the Spectral Problem for Periodic Elliptic Operators with Sign-Changing Density Function

Cited by 12 publications

References 14 publications

A Lyapunov type inequality for indefinite weights and eigenvalue homogenization

A Lyapunov type inequality for indefinite weights and eigenvalue homogenization

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

Spectrum of a diffusion operator with coefficient changing sign over a small inclusion

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