Some Results on the Schiffer's Conjecture in R^2

TL;DR

This paper investigates conditions under which a domain in the plane must be a disk, based on properties of Neumann eigenfunctions and eigenvalues, providing new partial results related to Schiffer's conjecture.

Contribution

It establishes new criteria involving Neumann eigenvalues and symmetry conditions that guarantee a domain is a disk, advancing understanding of Schiffer's conjecture in two dimensions.

Findings

Domains with certain Neumann eigenfunction boundary conditions are disks.

Convex, centrally symmetric domains with specific eigenvalue bounds are disks.

Provides partial affirmative results towards Schiffer's conjecture.

Abstract

Let $\Omega$ be an open, bounded domain in the plane with connected and smooth boundary, and $\omega$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $\mu > 0$. If the boundary value of $\omega$ is a nonzero constant along the boundary, denoting $0 = \mu_1(\Omega) < \mu_2(\Omega) <= ...$ the set of all Neumann eigenvalues for the Laplacian on $\Omega$, we show that 1) if $\mu < \mu_8(\Omega)$; or 2) if $\Omega$ is strictly convex and centrally symmetric, $\mu < \mu_13(\Omega)$, then $\Omega$ must be a disk.