On certain surface integrals related to the conormal derivative problem

The non-homogeneous conormal derivative problems for nonlinear, second-order divergence form elliptic equations with singular data appear naturally in mathematical modeling of real phenomena involving problems of image recovery, the thermistor problem, or studies of non-Newtonian fluids. We prove suitable estimates for certain surface integrals, related to non-homogeneous conormal derivative problems, which lead to essential boundedness of the weak solutions under quite general hypotheses on the data.