Global regularity in Lorentz spaces for nonlinear elliptic equations with Lp(⋠)log⁡L-growth

We prove a global Calderón-Zygmund-type estimate in the framework of Lorentz spaces for the gradients of weak solutions of nonlinear elliptic equations with Lp(⋅)log⁡L-growth. More precisely, by considering the zero-Dirichlet problem of the elliptic equationdiv A(x,Du)=div H(x,|F|)x∈Ω with A(x,ξ)=Dξ(a(x)|ξ|p(x)log⁡(e+|ξ|)) for any ξ≠0, A(x,0)=0, and |H(x,ξ)|≲(|F|p(x)−1log⁡(e+|F|)), we obtainΦ(x,|F|)∈L(t,q)(Ω)⇒Φ(x,|Du|)∈L(t,q)(Ω)fort>1andq∈(0,+∞], where Φ(x,|ξ|)=|ξ|p(x)log⁡(e+|ξ|), under the assumptions that the function a(⋅) has a small bounded mean oscillation seminorm, p(x) satisfies strong log-Hölder continuity, and the underlying domain is Reifenberg flat.