On the homogenization of degenerate elliptic equations in divergence form (Q1124044)

This paper deals with degenerate elliptic (nonlinear) Dirichlet problems \[ -\text{div}(a(x/\epsilon,Du_{\epsilon}))=f(x)\text{ in }\Omega,\quad u_{\epsilon}=0\text{ on }\partial \Omega, \] where \(\Omega\) is a bounded domain of \({\mathbb{R}}^ n\), and \(a(y,\xi)\) is a Carathéodory function, periodic in y, strictly monotone in \(\xi\) and such that \[ (C)\quad \lambda(y)| \xi |^ p\leq a(y,\xi)\cdot \xi,\quad | a(y,\xi)| \leq L\lambda(y)| \xi |^{p-1} \] for some \(p>1\) and \(\lambda(y)>0.\) The crucial assumption is that the weight function \(\lambda(y)\) belongs to the Muckenhoupt class \(A_ p\), i.e. for some positive constant K, \[ (\oint_{Q}\lambda dy)(\oint_{Q}\lambda^{-1/(p-1)}dy)^{p-1}\leq K \text{ for all cubes } Q \text{ of } {\mathbb{R}}^ n. \] Under these assumptions, we prove that the solutions \(\{u_{\epsilon}\}\) are converging in \(L^ 1(\Omega)\), as \(\epsilon\) tends to zero, to the solution \(u_ 0\) of \[ -div(b(Du_ 0))=f(x)\text{ in } \Omega,\quad u_ 0=0\text{ on } \partial \Omega, \] where \(b(\xi)\) is a certain continuous function satisfying some coerciveness and growth conditions like (C), with a positive constant \(\lambda_ 0\) in place of \(\lambda(y)\). Moreover the \(L^ 1(\Omega)\)-weak convergence of the momenta \(a(x/\epsilon,Du_{\epsilon})\) to \(b(Du_ 0)\) is proved. This result is an extension (to a special class of degenerate elliptic equations) of some classical results concerning the homogenization for monotone uniformly elliptic operators.