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.
Categories: math.AP