Homogenization of degenerate elliptic equations

We consider the divergent elliptic equations whose weight function and its inverse are assumed locally integrable. The equations of this type exhibit the Lavrentiev phenomenon, the nonuniqueness of weak solutions, as well as other surprising consequences. We classify the weak solutions of degenerate elliptic equations and show the attainability of the so-called $W$-solutions. Investigating the homogenization of arbitrary attainable solutions, we find their different asymptotic behavior. Under the assumption of the higher integrability of the weight function we estimate the difference between the exact solution and certain special approximations.