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 Hölder 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

of

{\mathbb R}^n R n

, local differentiability and partial Hölder continuity of the weak solutions

u u

of nonlinear elliptic systems of order

2m 2 m

in divergence form

\sum_{|\alpha|\leq m}(-1)^{|\alpha|} D^\alpha \, a^\alpha (x, Du) = 0. ∑ ∣ α ∣ ≤ m ( − 1 ) ∣ α ∣ D α   a α ( x , D u ) = 0.

Specifically, we generalize the results obtained by S. Campanato and P. Cannarsa [Differentiability and partial Hölder continuity of the solutions of nonlinear elliptic systems of order

2m 2 m

with quadratic growth, Ann. Scuola Norm. Sup. Pisa (4)8 (1981) 285–309] under the hypothesis that the coefficients

a^\alpha (x, Du) a α ( x , D u )

are strictly monotone with nonlinearity

q = 2 q = 2

.