Calderón–Zygmund estimates and non-uniformly elliptic operators

We consider a class of non-uniformly nonlinear elliptic equations whose model is given by -div(vertical bar Du vertical bar(p-2)Du a(x) vertical bar Du vertical bar(q-2)Du) = -div(vertical bar F vertical bar Fp-2 a(x)vertical bar F vertical bar Fq-2) where p < q and a(x) >= 0, and establish the related nonlinear Calderon-Zygmund theory. In particular, we provide sharp conditions under which the natural, and optimal, Calderon-Zygmund type result (vertical bar F vertical bar(p) + a(x)vertical bar F vertical bar(q)) is an element of L-loc(gamma) double right arrow (vertical bar Du vertical bar(p) + a(x) vertical bar Du vertical bar(q)) is an element of L-loc(gamma) holds for every gamma >= 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. (C) 2015 Elsevier Inc. All rights reserved.