On the Serrin problem for ring-shaped domains

TL;DR

This paper investigates the conditions under which solutions to a fluid flow problem in ring-shaped domains are rotationally symmetric, revealing that additional geometric constraints are necessary beyond classical boundary conditions.

Contribution

It demonstrates that constant Neumann boundary data alone do not guarantee symmetry, identifying the number of maximum points as a key condition, and introduces new comparison geometry methods.

Findings

Rotational symmetry requires a specific number of maximum points.

Constant wall shear stress implies finitely many maximal velocity streamlines.

Classical overdetermining conditions are insufficient for symmetry in ring domains.

Abstract

In this paper, we deal with the long standing open problem of characterising rotationally symmetric solutions to $\Delta u = -2$, when Dirichlet boundary conditions are imposed on a ring-shaped planar domain. From a physical perspective, the solution represents the velocity of a homogeneous incompressible fluid, flowing in steady parallel streamlines through a hollow cylindrical pipe and obeying a no-slip condition. In contrast with Serrin's classical result, we show that the simplest possible set of overdetermining conditions, namely the prescription of locally constant Neumann boundary data, is not sufficient to obtain a complete characterisation of the solutions. A further requirement on the number of maximum points arises in our analysis as a necessary and sufficient condition for the rotational symmetry. In fluid-dynamical terms, our results imply that if the wall shear stress is constant on some connected component of the pipe's wall, then the velocity of the fluid must attain its maximal value only at finitely many streamlines, unless the hollow pipe itself consists of a couple of concentric cylindrical round tubes. A major difficulty in the analysis of this problem comes from the lack of monotonicity of the model solutions, which makes the moving plane method ineffective. To remedy this issue, we introduce some new arguments in the spirit of comparison geometry, that we believe of independent interest.