  Second-order estimates are established for solutions to the $p$-Laplace
system with right-hand side in $L^2$. The nonlinear expression of the gradient
under the divergence operator is shown to belong to $W^{1,2}$, and hence to
enjoy the best possible degree of regularity. Moreover, its norm in $W^{1,2}$
is proved to be equivalent to the norm of the right-hand side in $L^2$. Our
global results apply to solutions to both Dirichlet and Neumann problems, and
entail minimal regularity of the boundary of the domain. In particular, our
conclusions hold for arbitrary bounded convex domains. Local estimates for
local solutions are provided as well.
