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

Bérard, Pierre; Helffer, Bernard

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 3, pp. 429-462.

Dans cet article, nous montrons que la « propriété étendue de Courant » est fausse pour certains domaines convexes lisses avec condition au bord de Neumann  : il existe une combinaison linéaire d’une première et d’une seconde fonctions propres de Neumann ayant trois domaines nodaux. Pour la démonstration, nous reformulons un argument de Jerison et Nadirashvili (J. Am. Math. Soc. 13 (2000)). Cet argument étant intéressant en lui-même, nous détaillons la preuve. En particulier, nous explicitons la dépendance des constantes par rapport à la géométrie des domaines lipschitziens le long des déformations.

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. Am. Math. Soc. 13 (2000)). 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.

Classification : 35P99, 35Q99, 58J50
Mots clés : Eigenfunction, Nodal domain, Courant nodal domain theorem

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