Given a bounded Euclidean domain $\Omega$, we consider the sequence of optimisers of the $k$^th Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $\Omega$ with fixed total volume. We show that this sequence encodes information yielding conditions for $\Omega$ to satisfy Pólya’s conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether Pólya’s conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Pólya’s original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa’s bound in terms of packing density.

中文翻译：

---

域和Pólya猜想的按比例缩放副本的最优并集

给定一个有界的欧几里得域$ \ Omega $，我们考虑该族中$ k $^个Laplacian特征值的优化子序列，该序列由所有可能的不固定的$ \ Omega $缩放副本的并集组成。我们表明，该序列编码信息生成条件\\ Omega $，以满足Dirichlet或Neumann边界条件下Pólya的猜想。这是Colbois和El Soufi结果的扩展，它仅适用于其中的域的系列包括的情况下，所有有界域。此外，我们根据Pólya的猜想是否适用于给定的特定域，对此类序列的不同可能行为进行了完全分类。通过这种方法，我们可以为满足某些动态台球条件的平铺域恢复Pólya原始结果的更强版本，并在堆积密度方面加强Urakawa的边界。