Towards a two-scale calculus

Abstract

Abstract. We define and characterize weak and strong two-scale convergence in L p , C 0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.Mathematics Subject Classification.

Homogenization of a Wilson–Cowan model for neural fields

“…Theorem 2). It is to be noted that Theorem 2 generalizes to the case of algebras with mean value, its counterpart proved by Visintin [29] in the special context of the algebras of continuous periodic functions. This result, based on the so-called sigma convergence concept, has allowed us to efficiently upscale a heterogeneous Wilson-Cowan type of models for neural fields.…”

“…The above result was first proved by Visintin [29] in the periodic setting by using the two-scale transform or unfolding method. Theorem 2 allows us to pass to the limit in the convolution terms without using neither the Fourier transform, nor the Laplace transform, and hence without restricting ourselves to the Hilbertian setting as it is the case in [30].…”

Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

“…The periodic unfolding technique, introduced in [9], allows for a natural definition of strong two-scale convergence and, hence, the treatment of nonlinear problems, cf. [10,11,12,13,14,15,16]. Based on this strong notion of convergence, one can ask for quantitative error estimates, see e.g.…”

A note on two scale compactness for $p = 1$

“…A few authors have touched on the problem, including Holmbom, Silfver, Svanstedt and Wellander in [16] and A. Visintin in [19], although detailed arguments seem to be unavailable in the literature. In [5,6] the authors address a related case of two-scale convergence in generalized Besicovitch spaces where there is also lack of separability.…”

“…The second makes use of the two-scale compactness proved for Radon measures by M. Amar in [2]. The last approach relies on the periodic unfolding characterization of two-scale limits, as introduced in [7] (see also [9,10,19]). The latter proof is the simplest and most intuitive, due to the fact that the periodic unfolding method reduces two-scale convergence in L p (Ω) to standard weak L p convergence in Ω×Y (where Y is the period of the oscillations) of the unfolded functions, thus allowing us to replace rapidly oscillating test functions with non-oscillatory test functions.…”