SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3098-3122, January 2021.
We prove norm-resolvent and spectral convergence in $L^2$ of solutions to the Neumann Poisson problem $-\Delta u_\varepsilon = f$ on a domain $\Omega_\varepsilon$ perforated by Dirichlet holes and shrinking to a 1-dimensional interval. The limit $u$ satisfies an equation of the type $-u''+\mu u = f$ on the interval $(0,1)$, where $\mu$ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighborhood and the vertex neighborhood is chosen correctly, the constant $\mu$ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation.

中文翻译：

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来自无处的奇怪顶点条件

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3098-3122 页，2021 年 1 月。
我们证明了诺依曼泊松问题 $-\Delta u_\varepsilon = f$ 在被狄利克雷孔穿孔并缩小到 1- 的域 $\Omega_\varepsilon$ 上的 $L^2$ 的范数解析和谱收敛性维区间。极限 $u$ 在区间 $(0,1)$ 上满足 $-u''+\mu u = f$ 类型的方程，其中 $\mu$ 是一个正常数。作为一种应用，我们研究了多孔图状域中解的收敛性。我们表明，如果边缘邻域和顶点邻域之间的缩放比例选择正确，则常数 $\mu$ 将出现在极限问题的顶点条件中。特别是，这意味着所产生的量子图的光谱通过穿孔以受控方式改变。