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

TL;DR

This paper establishes symmetry and monotonicity properties for solutions of vectorial p-Stokes systems using comparison principles and the moving-planes method, marking a novel contribution in the analysis of vectorial operators.

Contribution

It introduces the first qualitative results for vectorial p-Laplacian systems, extending symmetry and monotonicity analysis techniques to vector-valued functions.

Findings

Solutions are symmetric under certain conditions.

Solutions exhibit monotonicity properties.

The methods extend to vectorial operators for the first time.

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.