Elliptic homogenization with almost translation-invariant coefficients

Article type: Research Article

Authors: Goudey, Rémi^{; *}

Affiliations: École des Ponts ParisTech and INRIA Paris, 6 & 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France

Correspondence: [*] Corresponding author. E-mail: remi.goudey@enpc.fr.

Abstract: We consider an homogenization problem for the second order elliptic equation −div(a(·/ε)∇uε)=f when the coefficient a is almost translation-invariant at infinity and models a geometry close to a periodic geometry. This geometry is characterized by a particular discrete gradient of the coefficient a that belongs to a Lebesgue space Lp(Rd) for p∈[1,+∞[. When p<d, we establish a discrete adaptation of the Gagliardo–Nirenberg–Sobolev inequality in order to show that the coefficient a actually belongs to a certain class of periodic coefficients perturbed by a local defect. We next prove the existence of a corrector and we identify the homogenized limit of uε. When p⩾d, we exhibit admissible coefficients a such that uε possesses different subsequences that converge to different limits in L2.

Keywords: Homogenization, elliptic PDEs, corrector equation

DOI: 10.3233/ASY-221789

Journal: Asymptotic Analysis, vol. 132, no. 1-2, pp. 175-216, 2023