Global gradient estimates for the p(⋅)p(⋅)-Laplacian

We consider Calderón–Zygmund type estimates for the non-homogeneous p(⋅)p(⋅)-Laplacian system −div(|Du|p(⋅)−2Du)=−div(|G|p(⋅)−2G), where pp is a variable exponent. We show that |G|p(⋅)∈Lq(Rn)∩L1(Rn)|G|p(⋅)∈Lq(Rn)∩L1(Rn) implies |Du|p(⋅)∈Lq(Rn)∩L1(Rn)|Du|p(⋅)∈Lq(Rn)∩L1(Rn) for any q≥1q≥1. We also prove local estimates independent of the size of the domain and introduce new techniques to variable analysis. The paper is an extension of the local estimates of Acerbi–Mingione (2005).