Homogenization of Stiff Plates and Two-Dimensional High-Viscosity Stokes Equations

The paper deals with the homogenization of stiff heterogeneous plates. Assuming that the coefficients are equi-bounded in L ¹, we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients. This compactness result is based on a div-curl lemma for fourth-order equations. On the other hand, using an intermediate stream function we deduce from the plates case a similar result for high-viscosity Stokes equations in dimension two, so that the nature of the Stokes equation is preserved in the homogenization process. Finally, we show that the L ¹-boundedness assumption cannot be relaxed. Indeed, in the case of the Stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem, which violates the compactness result obtained under the L ¹-boundedness condition.