The publication of the important work of Rauch and Taylor [J. Funct. Anal. 18 (1975)] started a hole branch of research on wild perturbations of the Laplace–Beltrami operator. Here, we extend certain results and show norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbed Laplacian, i.e., the obstacles vanish. In the Dirichlet case, we consider two cases here: if the balls are too sparse, the limit operator is again the unperturbed one, while if the balls concentrate at a certain region (they become “solid” there), the limit operator is the Dirichlet Laplacian on the complement of the solid region. Norm resolvent convergence in the limit case of homogenisation is treated by Khrabustovskyi and the second author in another article (see also the references therein). Our work is based on a norm convergence result for operators acting in varying Hilbert spaces described in a book from 2012 by the second author.

中文翻译：

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剧烈扰动的流形：范数解析和谱收敛

劳赫和泰勒的重要著作出版[J．功能 肛门 [18（1975）]开始了对Laplace-Beltrami算子的野生扰动的研究。在这里，我们扩展某些结果并显示规范解决方案的趋同。我们考虑一个（不一定是紧凑的）歧管，其中去除了许多小球，随着半径的缩小，球的数量会增加，球的数量也可能是无限的。如果球的距离收缩的速度小于半径的收缩速度，则表明诺伊曼拉普拉斯算子收敛到不受干扰的拉普拉斯算子，即障碍消失了。在Dirichlet情况下，我们在这里考虑两种情况：如果球太稀疏​​，则限制算子再次是不受干扰的；如果球集中在某个区域（它们在该区域变为“实心”），则限制算子是Dirichlet拉普拉斯算子在实心区域的补码上。Khrabustovskyi和第二作者在另一篇文章中（也参见其中的参考文献）讨论了均质化极限情况下的规范解析收敛。