Symmetry and monotonicity results for solutions of vectorial $p$-Stokes systems

Abstract

In this paper we shall study qualitative properties of a $p$-Stokes type system, namely $$ -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $\Omega$},$$ where ${\boldsymbol \Delta}_p$ is the $p$-Laplacian vectorial operator. More precisely, under suitable assumptions on the domain $\Omega$ and the function $\boldsymbol{ f}$, it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to proceed with the moving-planes technique for systems. As far as we know, these are the first qualitative kind results involving vectorial operators.