c  Birkh¨auser Verlag, Basel, 2001 NoDEA Nonlinear differ. equ. appl. 00 (2001) 00–00 1021–9722/01/00000–00 $ 1.50+0.20/0 Homogenization for strongly anisotropic nonlinear elliptic equations ∗ Bruno FRANCHI † , Maria Carla TESI ‡ Dipartimento di Matematica Piazza di Porta S. Donato, 5 40127 Bologna, Italy e-mail: franchib@dm.unibo.it; tesi@dm.unibo.it Abstract. In this paper we present homogenization results for elliptic degen- erate differential equations describing strongly anisotropic media. More pre- cisely, we study the limit as ǫ → 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:  −div (α( x ǫ , ∇u)A( x ǫ )∇u)= f (x) ∈ L ∞ (Ω) u = 0 su ∂Ω where p> 1, α : R n × R n → R, α(y,ξ) ≈〈A(y)ξ,ξ〉 p/2-1 ,A ∈ M n×n (R), A being a measurable periodic matrix such that A t (x)= A(x) ≥ 0 almost everywhere. The anisotropy of the medium is described by the following structure hypotheses on the matrix A: λ 2/p (x)|ξ| 2 ≤〈A(x)ξ,ξ〉≤ Λ 2/p (x)|ξ| 2 , where the weight functions λ and Λ (satisfying suitable summability assump- tions) can vanish or blow up, and can also be ‘‘moderately” different. The convergence to the homogenized problem is obtained by a classical compen- sated compactness argument, that had to be extended to two-weight Sobolev spaces. Mathematics Subject Classification: 35J70. 35B40. Key words: Homogenization, weighted Sobolev spaces, compensated compactness. * Investigation supported by University of Bologna. Funds for selected research topics † Partially supported by MURST, Italy, and by GNAFA of INdAM, Italy. ‡ Supported by a post-doctoral joint grant of MURST and University of Bologna.