Level sets of certain Neumann eigenfunctions under deformation of Lipschitz domains. Application to the Extended Courant Property

TL;DR

This paper demonstrates that the Extended Courant Property does not hold for certain convex domains with Neumann boundary conditions by analyzing eigenfunction level sets under domain deformations.

Contribution

It provides a detailed deformation argument showing failure of the Extended Courant Property for specific smooth convex domains with Neumann conditions.

Findings

Extended Courant Property fails for some convex domains

Constructs linear combinations of eigenfunctions with three nodal domains

Provides detailed deformation analysis controlling geometric dependencies

Abstract

In this paper, we prove that the Extended Courant Property fails to be true for certain smooth, strictly convex domains with Neumann boundary condition: there exists a linear combination of a second and a first Neumann eigenfunctions, with three nodal domains. For the proof, we revisit a deformation argument of Jerison and Nadirashvili (J. Amer. Math. Soc. 2000, vol. 13). This argument being interesting in itself, we give full details. In particular, we carefully control the dependence of the constants on the geometry of our Lipschitz domains along the deformations.