We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser–Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side lengths. We also discuss applications of our results to Courant-sharp eigenfunctions and spectral partitioning.

中文翻译：

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第二Dirichlet本征函数的节点线估计

我们研究了广义曲线四边形中低能Dirichlet本征函数的节点曲线。这些技术可以看作是Grieser-Jerison在一系列凸面平面区域和具有一个弯曲边缘和较大长宽比的矩形的工作中开发的工具的概括。在这里，我们更详细地研究节点曲线的结构，因为我们找到了其曲率的精确边界，并对其直到直角与畴相交的两个点进行了统一的估计，并表明我们的许多结果都适用边长的纵横比相对较小。我们还将讨论我们的结果在库仑特锐特征函数和频谱划分中的应用。