Fractional Regularity for Nonlinear Elliptic Problems with Measure Data

We consider nonlinear elliptic equations of the type $$-\text{\rm div}\,a(x, Du)=\mu$$ having a Radon measure on the right-hand side and prove fractional differentiability results of Calder\'on-Zygmund type for very weak solutions. We extend some of the results achieved by G. Mingione [``The Calder\'on-Zygmund theory for elliptic problems with measure data'', Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6 (2007) 195--261], in turn improving a regularity result by G. R. Cirmi and S. Leonardi [``Higher differentiability for solutions of linear elliptic systems with measure data'', Discrete Contin. Dyn. Syst. 26 (2010) 89--104].