收缩管连接处本征函数的奇异性。部件I
Singularity of eigenfunctions at the junction of shrinking tubes. Part I
摘要：
考虑由一个细管连接的两个区域：可以证明，一般地，狄利克雷-拉普拉斯算子的给定本征函数的质量只集中在其中一个区域。当适当地归一化时，对另一域的限制在管的接合处产生奇点，因为通道截面趋于零。我们的主要结果表明，在非简并条件下，归一化极限轮廓具有N-1阶奇异性，其中N是空间维数。我们给出了本征函数在奇异结处的渐近行为的精确描述，这为我们提供了关于其在隧道入口附近的符号的一些重要信息。更准确地说，解决方案是一个符号的奇异交界处附近。换句话说，我们证明了节点集不进入通道内部。
英文摘要：
Consider two domains connected by a thin tube: it can be shown that, generically, the mass of a given eigenfunction of the Dirichlet Laplacian concentrates in only one of them. The restriction to the other domain, when suitably normalized, develops a singularity at the junction of the tube, as the channel section tends to zero. Our main result states that, under a nondegeneracy condition, the normalized limiting profile has a singularity of order N-1, where N is the space dimension. We give a precise description of the asymptotic behavior of eigenfunctions at the singular junction, which provides us with some important information about its sign near the tunnel entrance. More precisely, the solution is shown to be one-sign in a neighborhood of the singular junction. In other words, we prove that the nodal set does not enter inside the channel.
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