Oberseminar Mathematik in den Naturwissenschaften: Arianna Giunti, Homogenization in randomly perforated domains

Abstract:

We consider the homogenization of Poisson and Stokes equations in a bounded domain of R ^d ,
d > 2, perforated by many small random holes. We assume that the holes are generated by
suitably rescaled balls having random radii and centers (i.e. Boolean process). Our main as-
sumption is that the random radii of the Boolean process have finite (d − 2)-moment: This
condition is minimal in order to ensure that the average density of capacity generated by the
holes is finite, but still allows for the onset of clustering balls with overwhelming probability.
By combining analytic and probabilistic, percolation-like methods, we provide homogenization
results for a large class of measures as above.
This talk is based on joint works with R.M. Hofer and J.J.L. Velazquez (University of Bonn).