Differentiabilty and Partial Hölder Continuity of Solutions of Nonlinear Elliptic Systems

The authors continue the study of regularity properties for solutions of elliptic systems started by M. A. Ragusa [(1) Local H\"older regularity for solutions of elliptic systems, Duke Mathematical Journal 113 (2002) 385--397; (2) Continuity of the derivatives of solutions related to elliptic equations, Proc. Royal Society of Edinburgh 136(A) (2006) 1027--1039], proving, in a bounded open set $\Omega$ of ${\mathbb R}^n$, local differentiability and partial H\"older continuity of the weak solutions $u$ of nonlinear elliptic systems of order $2m$ in divergence form $$\sum_{|\alpha|\leq m}(-1)^{|\alpha|} D^\alpha \, a^\alpha (x, Du) = 0.$$ Specifically, we generalize the results obtained by S. Campanato and P. Cannarsa [Differentiability and partial H\"older continuity of the solutions of nonlinear elliptic systems of order $2m$ with quadratic growth, Ann. Scuola Norm. Sup. Pisa (4)8 (1981) 285--309] under the hypothesis that the coefficients $a^\alpha (x, Du)$ are strictly monotone with nonlinearity $q = 2$.