Regularity results for nonlinear elliptic systems

We prove partial regularity results for solutions to systems of elliptic partial differential equations with divergence structure, under nonstandard growth conditions. We consider solutions to −div a(x, u,Du) = b(x, u,Du), and use the method of A-harmonic approximation to show C1,α regularity almost-everywhere. We further calculate the optimal exponent, provided the operator a has Hölder continuous coefficients. We then relax the continuity assumption to allow for VMO or small BMO coefficients, and while a loss of regularity in the solution is to be expected, we retain almost-everywhere C0,α regularity in the solution. We then modify a technique of Campanato to further reduce the Hausdorff dimension of the singular set, assuming restrictions on the exponent and ambient dimension.