关于环形区域的Serrin问题
On the Serrin problem for ring-shaped domains
摘要：
本文讨论了当Dirichlet边界条件施加在环形平面区域上时，$Δu = -2$的旋转对称解的特征问题.从物理的角度来看，该解决方案代表了均匀不可压缩流体的速度，在稳定的平行流线通过中空圆柱形管道，并遵守无滑移条件。与Serrin的经典结果相比，我们表明，最简单的可能的一组超定条件，即处方的局部常数诺依曼边界数据，是不足以获得一个完整的表征的解决方案。作为旋转对称的充要条件，在我们的分析中出现了对极大点数目的进一步要求。在流体动力学方面，我们的结果意味着，如果壁面剪应力是恒定的一些连接组件的管壁，那么流体的速度必须达到其最大值只有在许多流线，除非空心管本身由一对同心圆柱形的圆管。分析该问题的一个主要困难是模型解缺乏单调性，这使得移动平面法无效。为了解决这个问题，我们引入了一些新的参数的比较几何的精神，我们相信独立的利益。
英文摘要：
In this paper, we deal with the long standing open problem of characterising rotationally symmetric solutions to $Δu = -2$, 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.
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