Calderón-Zygmund estimates and non-uniformly elliptic operators

We consider a class of non-uniformly nonlinear elliptic equations whose model is given by where p<q and a(x)≥0, and establish the related nonlinear Caldeon-Zygmund theory. In particular, we provide sharp conditions under which the natural, and optimal, Calderon-Zygmund type result. holds for every γ ≥ 1 These problems naturally emerge as Euler-Lagrange equations of some variational integrals introduced and studied by Marcellini [41] and Zhikov [53] in the framework of Homogenisation and Lavrentiev phenomenon.