On the global W^{2,q} regularity for nonlinear N-systems of the   p-Laplacian type in n space variables
  We consider the Dirichlet boundary value problem for nonlinear N-systems of partial differential equations with p-growth, 1<p<2, in the n-dimensional case. For clearness, we confine ourselves to a particularly representative case, the well known p-laplacian system. We are interested in regularity results, up to the boundary, for the second order derivatives of the solution. We prove W^{2,q}-global regularity results, for arbitrarily large values of q. In turn, the regularity achieved implies the Holder continuity of the gradient of the solution. It is worth noting that we cover the singular case \mu=0. See Theorem 2.1 below. 
H. Beirao da Veiga and F. Crispo
math.AP