$p$-Laplace Serrin problem on planar annuli

Abstract

We prove a rigidity result for the problem ${\Delta }_{p}u=-2$ defined on a ring-shaped domain in ${\mathbb{R}}^2$, when homogeneous Dirichlet boundary conditions are imposed and $p\in (1,2)$. This problem admits a natural physical interpretation, as it describes the steady laminar flow of a generalised Newtonian fluid exhibiting shear-thinning behaviour. Our symmetry result is obtained assuming that the set of maximum points, denoted by $\Sigma$, is a smooth closed curve, and there are no other critical points outside it; moreover, an appropriate asymptotic expansion of the solution near $\Sigma$ is required. Although our approach is inspired by the comparison strategy of [arXiv:2505.04458], the main ingredients used there are intrinsically linear and have no direct counterparts for the $p$-Laplacian. This requires the introduction of new ideas and techniques.