Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains

Article type: Research Article

Authors: Mel'nyk, Taras A. | Sivak, Olena A.

Affiliations: Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska St., Kyiv 01033, Ukraine. Tel.: +380 44 259 05 40; Fax: +380 44 259 07 85; E-mail: melnyk@imath.kiev.ua | Department of Mathematics, Swansea University, Swansea SA2 8PP, UK

Abstract: We consider quasilinear and linear boundary-value problems for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of order 𝒪(ε). The holes are divided into three ε-periodical sets depending on boundary conditions on their surfaces. On the boundaries of holes from one set we have the homogeneous Dirichlet conditions. On the boundaries of holes from the other sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For the solution to the quasilinear problem we find the leading terms of the asymptotics and prove the corresponding asymptotic estimates that show influence of the perturbation parameters. In the linear case we construct and justify the complete asymptotic expansion for the solution using two-scale asymptotic expansion method.

Keywords: perforated domain, asymptotic approximation and expansion, quasilinear and linear elliptic problems with rapidly oscillating coefficients

DOI: 10.3233/ASY-2011-1055

Journal: Asymptotic Analysis, vol. 75, no. 1-2, pp. 79-92, 2011