  We obtain some regularity results for solutions to vectorial $p$-Laplace
equations $$ -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf
div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol
u})\,\, \mbox{ in $\Omega$}\,.$$ More precisely we address the issue of second
order estimates for the stress field. As a consequence of our regularity
results we deduce a weighted Sobolev inequality that leads to weak comparison
principles. As a corollary we run over the moving plane technique to deduce
symmetry and monotonicity results for the solutions, under suitable
assumptions.
