Partial Regularity Result of Superquadratic Elliptic Systems with Dini Continuous Coefficients

Abstract

We consider the partial regularity for weak solutions to superquadratic elliptic systems with controllable growth condition, under the assumption of Dini continuous coefficients. The proof relies upon an iteration scheme of a decay estimate for a new type of excess functional. To establish the decay estimate, we use the technique of $\mathcal{A}$-harmonic approximation and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal Hölder exponent for the derivative of the weak solutions on the regular set.