University of Heidelberg, Applied Analysis Seminar
Thu 05.07.2018
14:15, posted in Applied Analysis
Dr. Arianna Giunti (Inst. Applied Mathematics, Bonn)
Homogenization in randomly perforated domains
INF 205, SR1
In the first part of the talk we present an homogenization result for solutions of a Poisson equation in a bounded domain of $\mathbf{R}^d$, $d > 2$, having many small holes. We assume that the holes have spherical shape and random radius and position. We show that for a class of stationary measures for the centres and radii of the holes, in the homogenized equation we obtain the averaged version of the "strange term" established by Cioranescu and Murat in the periodic case [D. Cioranescu and F. Murat, Un term étrange venu d'ailleurs (1986)]. We stress that in our setting we only require that the random radii of the holes have finite $(d - 2)$-moment: This condition is minimal in order to ensure that the average of the capacity of the holes is finite, but still allows for clustering phenomena with high probability. This is a joint work with R. Höfer and J.J.L. Velázquez (arXiv:1803.10214).

In the second part of the talk we turn to the analogue of the problem above in the case of a steady Stokes system. We show that in this case the solutions converge to the solution of an equation of Brinkmann type. This second part is an ongoing project with R. Höfer.

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