论文标题
光谱分区的稳定性和Dirichlet到Neumann地图
Stability of spectral partitions and the Dirichlet-to-Neumann map
论文作者
论文摘要
Laplacian征用功能的振荡提供了有关其定义的歧管的大量信息。该振荡可以在节点缺陷中编码，这是一个重要的几何量，众所周知，很难计算甚至估计。在这里，我们比较了两个最近获得的淋巴结缺陷的公式，一个是在歧管的电气空间上的能量功能，而另一个则根据在节点集上定义的双面dirichlet到neumann映射方面。我们通过根据Dirichlet到Neumann地图给予电气能量的Hessian的明确公式来联系这两种方法。这使我们能够根据相应的dirichlet到Neumann本征函数来计算Hessian本征函数，从而计算出最陡的下降的方向。我们的结果不假定两性，因此与光谱最小分区的研究有关。
The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy functional on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions.
