The concern in this paper is the problem of finding—or, at least, approximating—functions, defined within and on the boundary of a tangential polygon, functions whose Laplacian is $-1$ and which satisfy a homogeneous Robin boundary condition on the boundary. The parameter in the Robin condition is denoted by $\beta $. The integral of the solution over the interior, denoted by $Q$, is, in the context of flows in a microchannel, the volume flow rate. A variational estimate of the dependence of $Q$ on $\beta $ and the polygon’s geometry is studied. Classes of tangential polygons treated include regular polygons and triangles, especially isosceles: the variational estimate $R(\beta )$ is a rational function which approximates $Q(\beta )$ closely.

中文翻译：

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牛顿流体通过切向多边形微通道的稳定滑流

本文关注的问题是找到（或至少近似）在切线多边形的边界内和边界上定义的函数，这些函数的拉普拉斯算子为 $-1$ 并且满足边界上的齐次 Robin 边界条件. Robin 条件中的参数用 $\beta $ 表示。内部解的积分，用 $Q$ 表示，在微通道中的流动情况下，是体积流量。研究了$Q$ 对$\beta $ 和多边形几何形状的依赖性的变分估计。所处理的切线多边形类别包括正多边形和三角形，尤其是等腰：变分估计 $R(\beta )$ 是一个有理函数，它非常接近 $Q(\beta )$。