A convergence result for the periodic unfolding method related to fast diffusion on manifolds

Abstract

Based on the periodic unfolding method in periodic homogenization, we deduce a convergence result for gradients of functions defined on connected, smooth and periodic manifolds. Under the assumption of certain a-priori estimates of the gradient, which are typical for fast diffusion, the sum of a term involving a gradient with respect to the slow variable and one with respect to the fast variable is obtained in the homogenization limit. In addition, we show in a brief example how to apply this result and find for a reaction-diffusion equation defined on a periodic manifold that the homogenized equation contains a term describing macroscopic diffusion. RésuméA l'aide de la méthode d'éclatement périodique, nous démontrons un résultat de convergence des gradients de fonctions définies sur des variétés connexes, différentiables et périodiques. Sous certaines conditions d'estimation du gradient, typiques de la diffusion rapide, nous obtenonsà la limite d'homogénéisation la somme d'un gradient de la variable globale et d'un gradient de la variable locale. Un exemple illustre l'utilisation de ce résultat: pour uné equation de réaction et diffusion définie sur une variété périodique, nous démontrons que l'équation homogénéisée contient un terme décrivant une diffusion globale.

Diffusion on Surfaces and the Boundary Periodic Unfolding Operator with an Application to Carcinogenesis in Human Cells

“…It allows us to apply the boundary periodic unfolding operator for diffusion equations defined on smooth manifolds. For linear reaction-diffusion equations defined on manifolds it is also possible to use twoscale convergence for the homogenization process; see [23,2], and we also refer to [15] for results on fast diffusion on manifolds. But if there are nonlinear reaction terms in the equation, strong convergence of the functions typically is required.…”

Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces

“…Remark 6. If in Lemma 3.3 it additionally holds that u Γ 0 ∈ H 1 (Ω), then from the proof of Theorem 3.4 below, see also [14] for the case when Γ is connected, we get u → u Γ 0 in the two-scale sense on Γ…”

“…The assumptions of Corollary 2 are fulfilled, for example, if there exists an extensionũ ∈ H 1 (Ω) of u , such that the sequence (ũ ) is bounded in H 1 (Ω). This is always the case when the surface Γ is connected, see [14]. However, in Section 4 we show, that the H 1 -regularity of the two-scale limit is in general not guaranteed if Γ is disconnected, i. e., for suitable u ∈ H 1 (Γ ) with √ u H 1 (Γ ) ≤ C we show the…”

“…For sequences defined only on the oscillating surface Γ the convergence result (1) for the tangential gradient was already stated without proof in [4,15], however, the case of a disconnected surface when the result fails was not excluded explicitly. In [14], it was proven for functions on a connected surface Γ using the unfolding method together with an extension argument. In [5], the result was shown for the disconnect and connected domain using the unfolding operator under the additional assumption that the two-scale limit is an element of H 1 (Ω).…”

Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology

“…[14]). Since ∇ϕ ε j = ∇ x Ψ j + εψ j,ε ∇ x θ j + θ j (∇ ξ ψ j,ε ), and thanks to Proposition 2.8 in [14] (see also [26]), there holds…”