Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Calderon--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.