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

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.

References in corpus (5)

• Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle
• Sturm's theorem on zeros of linear combinations of eigenfunctions
• Nodal domains in the square---the Neumann case
• On Courant's nodal domain property for linear combinations of eigenfunctions, Part I
• A toy Neumann analogue of the nodal line conjecture