Global second order optimal regularity for the vectorial $p$-Laplacian

We obtain optimal regularity results for solutions to vectorial $p$-Laplace equations -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x)\,\, \mbox{ in $\Omega$}\,. More precisely we address the issue of global second order estimates for the stress field.