Several Questions Related to Homogenization
A point that this survey talk may illustrate is that one could sometime deduce from a weak convergence some remarkably strong a priori pointwise estimates. The latter may be viewed also as a uncertainty quantification. Through a couple concrete problems in highly oscillating medium, we lead to study several questions of classical nature: Liouville type theorems, structure of ancient solutions, uniform controllability and observability, quantitative unique continuation and absence of \(L^2\) eigenvalues and eigenfunction asymptotics.
Though the talk is centered at the periodic case, ideas could be applied to many other situations including large scale geometric analysis.