Gradient Estimates for Equations from Composite Materials under Orlicz Growth

We consider nonlinear elliptic equations arising from composite materials with nonstandard growth. The domain governed by nonlinearities with Orlicz growth consists of multiple subdomains representing different materials. We assume that the boundaries of the domain and its subdomains are sufficiently regular in the Reifenberg-flat sense. Furthermore, in each subdomain, the nonlinearities are locally merely measurable in one direction, and the oscillation of the nonlinearities in the other directions is assumed to be sufficiently small in the bounded mean oscillation (BMO) sense. Under these settings, we establish a global Calderón–Zygmund theory based on the geometric properties of Reifenberg flat domains and the nonlinear measurabilities.