Introverted algebras with mean value and applications

Abstract

Let A be an introverted algebra with mean value. We prove that its spectrum ∆(A) is a compact topological semigroup, and that the kernel K(∆(A)) of ∆(A) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K(∆(A)) is an ideal of ∆(A), we define the convolution over ∆(A). We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.

Multiscale nonlocal flow in a fractured porous medium

“…Proof. The proof is very similar to the one of its homologue Theorem 2.6 in [44] (see also [43]). Since Theorem 2.6 in [44] involves almost periodicity and moreover is checked in the two-scale sense, it is suitable to repeat the proof here in the periodicity and multiscale frameworks for completeness.…”

Homogenization of nonlinear parabolic equations with hysteresis

“…In this section we recall the main properties and some basic facts about the concept of sigma-convergence. We refer the reader to [22,23] for the details regarding most of the results of this section.…”

Sigma-convergence of semilinear stochastic wave equations

“…For u = v + N ∈ B p A (R N ) (1 ≤ p ≤ ∞) and y ∈ R N , we define in a natural way the translate τ y u = v(· + y) + N of u, and as it can be seen in [49,50], this is well defined and induces a strongly continuous N -parameter group of isometries T (y) :…”

“…We denote by ∂/∂y i (1 ≤ i ≤ N ) the infinitesimal generator of T (y) along the ith coordinate direction. We refer the reader to [49,50] for the properties of ∂/∂y i . Now, let…”