Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition

• Published in: Advances in nonlinear analysis Vol. 8; no. 1; pp. 679 - 693
• Main Authors: Díaz, Jesus Ildefonso, Gómez-Castro, David, Podol’skii, Alexander V., Shaposhnikova, Tatiana A.
• Format: Journal Article
• Language: English
• Published: De Gruyter 03.08.2017
• Subjects: 35B25; 35B40; 35J05; 35J20; Homogenization; noncritical sizes, maximal monotone graphs; nonlinear boundary reaction
• ISSN: 2191-9496; 2191-950X
• DOI: 10.1515/anona-2017-0140

The aim of this paper is to consider the asymptotic behavior of boundary value problems in -dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a -Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which , . In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.