Abstract for the talk on 03.11.2017 (10:00 h)Arbeitsgemeinschaft ANGEWANDTE ANALYSIS
Tuan Anh Nguyen (Universität Duisburg-Essen)
A Liouville property for the random conductance model
In the talk, I will give an introduction to my research based on a work by Bella, Fehrman, and Otto on stochastic homogenization. Consider the random diﬀerential operator ∇⋅ a∇ where the random matrix (coeﬃcient ﬁeld) a is assumed to be stationary and ergodic. By making use of the extended correctors (ϕ,σ) and choosing a reasonable homogenization error, they can obtain a regularity estimate, namely the excess decay, which implies a Liouville principle for a-harmonic functions, i.e. functions satisfying ∇⋅ a∇u = 0. It is interesting to know whether their ideas work in the discrete case (the random conductance model on the lattice). The answer is positive: By using several analytic and numerical methods, it is possible to implement their ideas in the discrete case.