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

Abstract

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.

Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

“…. This shows that (16) are actually necessary and sufficient conditions of self-adjointness. It takes elementary calculus to check that this is equivalent to what is announced in the statement of the proposition.…”

“…Now, let us consider an operator A which satisfies (16). Let us prove that A is a self-adjoint extension of A.…”

“…Instead, we will employ a method introduced in [30, Chap. 2], [33] (see also [15,16] for examples of application in the context of negative materials). Our strategy is as follows.…”

Team organization may help swarms of flies to become invisible in closed waveguides

“…In the above expression, the functions v 0, n , u 1 , v 1, 2 , u 2 , are respectively defined in (12), (15), (16), (18).…”

“…). As a consequence, the term u 1 defined by(15) indeed cancels the discrepancy2 n=1 [∆, ζ n ] + k 2 ζ n Id u 0 (M n ) cap(O) |x − M n |at order ε. Withv 1, 1 , we impose the homogeneous Dirichlet boundary condition on ∂O ε 1 (τ ) at order ε.…”