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. During the talk, following a paper in collaboration with Luigi Montoro and Berardino Sciunzi, we will obtain under suitable assumptions, sharp second order estimates, thus proving the optimal regularity of the vector field \(|\nabla u|^{p−2}\nabla u\), up to the boundary.