Prof. F. Otto, Max-Plank-Inst. in Leipzig:
Effective behavior of random media: From an error analysis to elliptic regularity theory
Zeit und Ort
Donnerstag, 2.2.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Zusammenfassung
Heterogeneous media, like a sediment, are often naturally described in statistical terms. \nHow to extract their effective behavior on large scales, like the permeability in Darcy's law, from the\nstatistical specifications? A practioners numerical approach is to sample the medium \naccording to these specifications and to determine\nthe permeability in the Cartesian directions by imposing simple boundary conditions.\nWhat is the error made in terms of the size of this "representative volume element''?\nOur interest in what is called "stochastic homogenization'' grew out of this error analysis.\n\nIn the course of developing such an error analysis, connections with the classical\nregularity theory for elliptic operators have emerged. It turns out that the\nrandomness, in conjunction with statistical homogeneity, of the coefficient field (which can be seen as a Riemannian metric)\ngenerates large-scale regularity of harmonic functions (w.r. t.the corresponding Laplace-Beltrami operator). \nThis is embodied by a hierarchy of Liouville properties: \nAlmost surely, the space of harmonic functions of given but arbitrary growth rate\nhas the same dimension as in the flat (i.e. Euclidean) case. \nClassical examples show that from a deterministic point of view, this Liouville property fails \nalready for a small growth rate:\nThere are (smooth) coefficient fields, which correspond to the geometry of a cone at infinity,\nthat allow for sublinearly growing but non-constant harmonic functions