Regularity and symmetry results for the vectorial p-Laplacian

Abstract

We obtain some regularity results for solutions to vectorial $p$-Laplace equations $-{\Delta }_{p}u=-\mathrm{div}\left({\left|Du\right|}^{p-2}Du\right)=f(x,u)\text{in}\Omega .$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.