论文标题
如何听到鼓的角落
How to hear the corners of a drum
论文作者
论文摘要
我们宣布一个新的结果表明，在Dirichlet，Neumann或Robin边界条件下，平面域中的角落是Laplacian的光谱不变。对于多边形域的情况，我们以KAC的“不感觉边界的原理”的精神来展示如何将局部原理与明确模型热内核的计算一起使用以证明结果。在此过程中，我们在所有三个边界条件下都证明了该位置原则。尽管以前以Dirichlet边界条件而闻名，但在此处介绍的一般性中，这似乎是Robin和Neumann边界条件的新事物。对于曲线多边形的情况，我们描述了使用局部原理的相同参数如何失败，但是可以用强大的微局部分析方法代替。
We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary conditions, in the generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle fail, but can nonetheless be replaced by powerful microlocal analysis methods.
