Gradient estimates for nonlinear elliptic equations with vanishing Neumann data in quasiconvex domains

We study the conormal derivative problem for an elliptic equation of p-Laplacian type with discontinuous coefficients in a non-smooth domain in order to look for the minimal assumptions necessary to have the nonlinear Calderón–Zygmund theory for such problem. Under the assumptions that the nonlinear operator is sufficiently close to the p  -Laplacian operator in BMO semi-norm and the boundary of the domain can be locally approximated by the convex boundary, we prove that both the gradient and the associated nonhomogeneous term belong to the same LqLq space for every q∈[p,∞)q∈[p,∞). As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction.