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

This paper is devoted to the homogenization of the problem −div(a_ε∇u_ε)+ν u_ε=f in a bounded domain Ω of R^d, with Neumann's (ν=1) or Dirichlet's (ν=0) boundary conditions. The conductivity matrix a_ε is defined by $a_{\varepsilon }(x):=A_{\varepsilon }(\tfrac{x}{\varepsilon })$ where (A_ε)_ε>0 is a sequence of bounded but non‐uniformly elliptic periodic matrix‐valued functions.

We make a general assumption on A_ε for that the sequence u_ε 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 A_ε uniformly looses its ellipticity as ε tends to zero. Finally we illustrate the optimality of our condition on A_ε in the framework of isolating thin layers.