Sharp gradient estimates for quasilinear elliptic equations with p(x) growth on nonsmooth domains

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the p(x)-Laplacian on nonsmooth domains and obtain sharp Calderón-Zygmund type estimates in the variable exponent setting. In a recent work of [12], the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p(x), see (1.3) and (1.4). Here, we bridge this gap to obtain the end point case of the estimates obtained in [12], see (1.5). In order to do this, we have to obtain significantly improved a priori estimates below p(x), which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature.