Nodal lengths of eigenfunctions in the disc

TL;DR

This paper establishes precise bounds on the lengths of nodal lines of Laplacian eigenfunctions in a disc and explores geometric properties of eigenfunctions with maximal nodal lengths.

Contribution

It provides the first sharp bounds for nodal lengths in the disc and analyzes the geometry of eigenfunctions achieving these bounds.

Findings

Sharp lower and upper bounds for nodal lengths derived.

Identification of geometric properties of eigenfunctions with maximal nodal length.

Insights into the structure of eigenfunctions in the disc.

Abstract

In this paper, we derive the sharp lower and upper bounds of nodal lengths of Laplacian eigenfunctions in the disc. Furthermore, we observe a geometric property of the eigenfunctions whose nodal curves maximize the nodal length.