Asymptotic Analysis

We present an extension of the two‐scale convergence method to the homogenization of monotone Dirichlet problems in periodically perforated domains, with critical size. The case of operators of order p strictly less than the dimension N of the space was considered in a previous paper. The case p=N presents several differences because it is a limit case for the Sobolev's imbedding theorem. Its study is the aim of this paper.

We give a semiclassical study of the spectrum of a two‐dimensional model of Harper's operator. We localize the part of the spectrum near the maximum in exponentially small intervals and give estimate on the splitting between them. Then we prove that in the rational case the Lebesgue measure of the spectrum remains strictly positive even in the semiclassical limit.

We consider a model developed by Savage and Hutter [17,18], which describes the flow of granular avalanches down a smoothly varying slope. The system consists of two nonstrictly hyperbolic equations for height and momentum.

The existence of entropy solutions to this model is proved using the vanishing viscosity method, where we make extensive use of a generalised version of the invariant region theorem in order to prove a priori estimates. Since the model has a discontinuous source term, a new definition of entropy solution must be introduced.

In the paper we propose a new approach to the homogenization theory on periodic wire‐networks and junctions, based on singular measures on these structures. We characterize the Sobolev spaces on such constructions and describe the fields of potential and solenoidal (divergence free) vector‐function. Then we compare the effective coefficients obtained for the singular structures and the classical effective coefficients for thin constructions with vanishing thickness, and show that the corresponding diagram is commutative.