  We consider the Dirichlet boundary value problem for nonlinear systems of
partial differential equations with p-structure. We choose two representative
cases: the "full gradient case", corresponding to a p-Laplacian, and the
"symmetric gradient case", arising from mathematical physics. The domain is
either the "cubic domain" (see the definition below) or a bounded open subset
of R^3 with a smooth boundary. We are interested in regularity results, up to
the boundary, for the second order derivatives of the velocity field. Depending
on the model and on the range of p, p<2 or p>2, we prove different regularity
results. It is worth noting that in the full gradient case with p<2 we cover
the degenerate case and obtain W^{2,q}-global regularity results, for
arbitrarily large values of q.
