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

We study a spectral problem $(\mathscr{P}^δ)$ 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 $(\mathscr{P}^δ)$ consists of two sequences converging to $\pm\infty$. However, at the limit $δ=0$, the small inclusion vanishes so that there should only remain positive spectrum for $(\mathscr{P}^δ)$. What happens to the negative spectrum? In this paper, we prove that the positive spectrum of $(\mathscr{P}^δ)$ tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of $(\mathscr{P}^δ)$ 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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