Optimal conditions of convergence and effects of anisotropy in the homogenization of non‐uniformly elliptic problems

Article type: Research Article

Authors: Briane, Marc

Affiliations: Laboratoire LANS, INSA de Rennes & IRMAR, 20, avenue des Buttes de Coësmes, CS 14315, 35043 Rennes cedex, France E‐mail: mbriane@insa‐rennes.fr

Abstract: This paper is devoted to the homogenization of the problem −div(aε∇uε)+ν uε=f in a bounded domain Ω of Rd, with Neumann's (ν=1) or Dirichlet's (ν=0) boundary conditions. The conductivity matrix aε is defined by $a_{\varepsilon }(x):=A_{\varepsilon }(\tfrac{x}{\varepsilon })$ where (Aε)ε>0 is a sequence of bounded but non‐uniformly elliptic periodic matrix‐valued functions. We make a general assumption on Aε for that the sequence uε strongly converges in L2(Ω) to a function u0 solution of a similar problem. We also yield an example in which the compactness result holds true although the sequence Aε uniformly looses its ellipticity as ε tends to zero. Finally we illustrate the optimality of our condition on Aε in the framework of isolating thin layers.

Journal: Asymptotic Analysis, vol. 25, no. 3-4, pp. 271-297, 2001