  Solutions to $p$-Laplace equations are not, in general, of class $C^2$. The
study of Sobolev regularity of the second derivatives is, therefore, a crucial
issue. An important contribution by Cianchi and Maz'ya shows that, if the
source term is in $L^2$, then the field $|\nabla u|^{p-2}\nabla u$ is in
$W^{1,2}$. The $L^2$-regularity of the source term is also a necessary
condition. Here, under suitable assumptions, we obtain sharp second order
estimates, thus proving the optimal regularity of the vector field $|\nabla
u|^{p-2}\nabla u$, up to the boundary.
