A boundary estimate for nonlinear equations with discontinuous coefficients

Abstract

We prove a boundary estimate for the gradient of a weak solution to a partial differential equation of $p$-Laplacian type with coefficients of vanishing mean oscillation. Combining the boundary estimate with a known interior estimate we obtain a global higher integrability result. The main result generalizes known results for linear equations to a nonlinear case. Our method is based on choosing the right test function, and, in particular, we do not use any representation formulas for solutions.