Regularity and symmetry results for the vectorial p-Laplacian

We obtain some regularity results for solutions to vectorial p-Laplace equations-Delta(p)u = -div (vertical bar Du vertical bar(p-2) Du) = f (x, u ) 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.