Homogenization of periodic differential operators as a spectral threshold effect
Seminar Room 1, Newton Institute
In L_2(R^d), we study matrix periodic elliptic second order differential operators of the form A=X^*X. Here X is a homogeneous first order differential operator. Many operators of mathematical physics can be written in this form. We study a homogenization problem for A. Namely, for the corresponding operator with rapidly oscillating coefficients, we study the behavior of the resolvent in the small period limit. We find approximation for this resolvent in the (L_2)-operator norm in terms of the resolvent of the effective operator. The error estimate is order-sharp, and the constant in this estimate is well controlled. Next, taking the so called corrector into account, we obtain more accurate approximations for the resolvent both in the (L_2)-operator norm and in the "energetic" norm. The error estimates are order-sharp. The obtained results are of new type in the homogenization theory.
The method is based on the abstract operator theory approach for selfadjoint operator families A(t)=X(t)^*X(t) depending on one-dimensional parameter t, where X(t)=X_0 +tX_1. It turns out that the homogenization procedure is determined by the spectral characteristics of the periodic operator A near the bottom of the spectrum. Therefore, a homogenization procedure can be treated as a spectral threshold effect.
General results are applied to particular periodic operators of mathematical physics: the acoustics operator, the operator of elasticity theory, the Maxwell operator, the Schrodinger operator, the twodimensional Pauli operator.