Approximation locale précisée dans des problèmes multi-échelles avec défauts localisés

Abstract

Présenté par £££££ RésuméNous poursuivons l'étude initiée dans [3] de problèmes multi-échelles avec défauts, dans le cadre de la théorie de l'homogénéisation, spécifiquement ici pour une équation de diffusion avec un coefficient de la forme fonction périodique perturbée par une fonction L r (R d ), 1 < r < +∞, modélisant un défaut local. Nous esquissons la démonstration du fait que le correcteur, dont l'existence a été prouvée dans [3,4], permet d'approcher la fonction solution de l'équation originale avec la même précision, essentiellement, que dans le cas purement périodique. Les taux de convergence varient, et sont précisés, en fonction de l'intégrabilité L r du défaut. Une extension à un cas abstrait "général" est mentionnée. Les résultats annoncés dans cette Note seront précisés dans les documents [2,12]. AbstractLocal precised approximation in multiscale problems with local defects.We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an L r (R d ) , 1 < r < +∞, function modeling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3,4], allows to approximate the solution of the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the L r integrability of the defect. The generalization to an abstract setting is mentioned. Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [13], and borrows a lot from it. The details of the results announced in this Note are given in our publications [2,12].

Deterministic homogenization of elliptic equations with lower order terms

“…We point out that, to our knowledge, the result in Theorem 1.3 is new and extends the results obtained in [3,4,5,6,14] to more general elliptic equations in at least two perspectives:…”

“…We also point out an important fact. With less regularity than in [5,6], however we obtain sharper L 2 convergence rates than in [5,6]. Indeed in [5,6] it is shown that if the matrix A has the form…”

“…). Theorem 1.2 has been proved in [3,4,5,6] (see also [14]) under the following restrictive assumption on the coefficients A and V :…”

“…2) The assumption (A1) is made essentially in order to obtain the boundedness of the gradient of the periodic corrector, which is a crucial step in [3,4,5,6,14]. For the existence of the asymptotic periodic corrector, we need the existence of its periodic component.…”

Some quantitative homogenization results in a simple case of interface

“…In the periodic setting, such results are provided by Avellaneda and Lin's theory [7]. But, as shown in [9] (see also [6,10]), the periodicity assumption is not necessary to these local estimates: they can be obtained in various frameworks, as long as the correctors and the potential (defined by (14) and ( 22)) associated with the matrix A are strictly sublinear and as long as the homogenized matrix is constant.…”