Optimal conditions of convergence and effects of anisotropy in the homogenization of non‐uniformly elliptic problems
Marc Briane · Asymptotic Analysis · 2001
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 L 2 (Ω) to a function u 0 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.