Optimal partial regularity for very weak solutions to a class of nonlinear elliptic systems

We consider optimal partial regularity for very weak solutions to a class of nonlinear elliptic systems and obtain the general criterion for a very weak solution to be regular in the neighborhood of a given point. First, by Hodge decomposition and the technique of filling holes, we establish the relation between the very weak solution and the classical weak solution. Furthermore, combining the technique of p-harmonic approximation with the method of Hodge decomposition, we obtain the partial regularity result. In particular, the partial regularity we obtained is optimal.