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We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type...

Reiterated homogenization is studied for divergence structure parabolic problems of the form $\partial u_{\epsilon }/\partial t-div\left(a\left(x,x/\epsilon ,x/{\epsilon }^2,t,t/{\epsilon }^k\right)\nabla u_{\epsilon }\right)=f$. It is shown that under standard assumptions on the function $a(x,y_1,y_2,t,\tau )$ the sequence $\left\{u_{ϵ}\right\}$ of solutions converges weakly in $L^2(0,T;H_0^1\left(\Omega \right))$ to the solution $u$ of the homogenized problem $\partial u/\partial t-div\left(b\right(x,t\left)\nabla u\right)=f$.

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in $L^2\left(\Omega \right)$ involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.