Andrey L. Piatnitski: Homogenization of elliptic spectral problems with sign changing (part 3)

The goal of the course is to present recent results on the asymptotic behaviour of spectra of second order self-adjoint elliptic operators with periodic rapidly oscillating coefficients and sign changing density function. We will consider the Dirichlet spectral problem in a regular bounded domain. First we will show that, for each fixed value of small parameter, the spectrum of this problem is discrete and consists of two infinite series, one of them converges to +∞, and another to −∞. The limit behaviour of spectrum depends crucially on whether the mean value of the weight function is positive, or negative, or equal to zero. We will construct the asymptotics of eigenpairs in all three cases. We will also discuss spectral problems of Steklov type and spectral problems for elliptic systems.