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On the homogenization of degenerate elliptic equations in divergence form. (English) Zbl 0678.35036
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.
Reviewer: F.Serra Cassano

35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J25 Boundary value problems for second-order elliptic equations