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We analyze the behavior of weak solutions to compressible viscous fluid flows in a bounded domain in R 3 , randomly perforated by tiny balls with random size. Assuming the radii of the balls scale like ε α , α > 3 , with ε denoting the average distance between the balls, the problem homogenize to the same limiting equation. Our main contribution is a construction of the Bogovskiĭ operator, uniformly in ε , without any assumptions on the minimal distance between the balls. [ABSTRACT FROM AUTHOR]

We study the transition from flat to wrinkled regions in a uniaxially stretched thin elastic film. We set up a model variational problem, and study the energy of its ground state. Using known scaling bounds for the minimal energy, the minimal energy can be written as a minimum of the underlying (convex) relaxed problem plus a term that grows linearly in the thickness of the film.We show that in the limit of vanishing thickness the prefactor in the scaling law for the original problem can be obtained by minimization of simpler scalar constrained variational problems. [ABSTRACT FROM AUTHOR]