Uniform estimates in two periodic homogenization problems

We analyze two partial differential equations that are posed on perforated domains. We provide a priori estimates that do not depend on the size of th e perforation: A sequence of solutions is uniformly bounded in a Sobolev sp ace of regular functions. The first homogenization problem concerns the Lap lace and the mean-curvature operator with Neumann boundary conditions. We d erive uniform Lipschitz estimates for the solutions. The result is used in the analysis of a free boundary system of fluid mechanics. A contractive it eration yields the existence of solutions and uniform estimates. The key is the use of function spaces that are different from the usual L-p-spaces. ( C) 2000 John Wiley & Sons, Inc.