Two scale defect measure and linear equations with oscillating coefficients

Abstract

Microlocal measure $\mu$ is associated to a two-scale convergent sequence $u_n$ over $\R^d$ with the limit $u\in L^2(\R^d\times \T^d),$$\T^d$ is a torus, to analyze possible strong limit. $\mu$ is an operator valued measure absolutely continuous with respect to the product of scalar microlocal defect measure and a measure on the $d$-dimensional torus. The result is applied to the first order linear PDE with the oscillating coefficients.