Homogenization of the Neumann problem with nonisolated holes

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Abstract
We consider the homogenization of second-order elliptic equations with a Neumann boundary condition in open sets periodically perforated with holes of the size of the period. When the holes are isolated, Cioranescu and Saint Jean Paulin (1979) proved the convergence of the homogenization process. One of their main tool was the construction of an extension of the solution, which is uniformly bounded. In the present paper, we give a new proof of the convergence, which avoids the use of such an extension. The main advantage of our approach is that it generalizes the result of Cioranescu and Saint Jean Paulin to the general case of periodic holes which may be not isolated (including, for example in three dimensions, the case of a domain perforated by interconnected cylinders).

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