JournalsjstVol. 11, No. 1pp. 323–353

Nodal line estimates for the second Dirichlet eigenfunction

  • Thomas Beck

    Fordham University, Bronx, USA
  • Yaiza Canzani

    University of North Carolina, Chapel Hill, USA
  • Jeremy L. Marzuola

    University of North Carolina, Chapel Hill, USA
Nodal line estimates for the second Dirichlet eigenfunction cover
Download PDF

This article is published open access under our Subscribe to Open model.

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.

Cite this article

Thomas Beck, Yaiza Canzani, Jeremy L. Marzuola, Nodal line estimates for the second Dirichlet eigenfunction. J. Spectr. Theory 11 (2021), no. 1, pp. 323–353

DOI 10.4171/JST/342