# Nodal line estimates for the second Dirichlet eigenfunction

## TL;DR

This paper analyzes the structure and curvature of nodal lines of low-energy Dirichlet eigenfunctions in curvilinear quadrilaterals, extending previous methods and providing precise bounds with applications to spectral theory.

## Contribution

It generalizes existing tools to study nodal curves in more complex domains, offering detailed curvature bounds and insights for small aspect ratios.

## Key findings

- Derived uniform curvature bounds for nodal curves
- Extended analysis to domains with small aspect ratios
- Discussed implications for Courant-sharp eigenfunctions

## Abstract

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.