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.
