Homogenization of foliated annuli

Let \(\Omega = \Omega _0 \backslash \bar \Omega _1\) be a regular annulus inR ^N and \(\phi :\bar \Omega \to R\) be a regular function such that φ=0 on ?Ω₀, φ=1 on ?Ω₁ and ▽φ ≠ 0. Let K_n be the subset of functions v ε W^1,p (Ω) such that v=0 on ?Ω₀, v=1 on ?Ω₁, v=(unprescribed) constant on n given level surfaces of φ. We study the convergence of sequences of minimization problems of the type $$Inf\left\{ {\int\limits_\Omega {\frac{1}{{a_n \circ \phi }}G(x,(a_n \circ \phi )\nabla v)dx;v \in K_n } } \right\},$$ where a_n ε L^∞ (0,1) and G: (x, ζ) ε Ω × R^N → G(x, ζ εR is convex with respect to ξ and verifies some standard growth conditions.