Organizers: Thomas Eiter, Marita Thomas, Martin Heida

Abstract:

In many problems from material science, geophysics and fluid dynamics, certain material or geometrical properties vary strongly on a local scale. To study their effect on the macroscopic behavior and to obtain effective models, one can derive averaged equations by passage to a homogenized framework. Moreover, by introducing some randomness, one can take into account that the exact distribution of the small-scale fluctuations is usually unknown in applications. This minisymposium brings together experts on the mathematical investigation of such stochastic homogenization problems

Speakers:

Peter Bella

Stefan Neukamm

Caterina Zeppieri

Abstract:

Nonlocal energies, such as fractional Sobolev seminorms, arise naturally in mathematical models involving long-range interactions. In this talk, we study minimizers of such energies that vanish on a collection of small balls with random centers and radii, leading to a bilateral (fractional) obstacle problem. I will present a homogenization result that holds under minimal assumptions on the distribution and size of the obstacles, which are generated by a stationary marked point process. In particular, the obstacles may overlap and form clusters with positive probability, giving rise to a complex microstructure. Our analysis identifies the limiting energy and shows how it reflects the underlying probability distribution of the obstacles.