Courant-sharp Robin eigenvalues for the square and other planar domains

K Gittins (1) , Bernard Helffer (2)

1 UNINE - Université de Neuchâtel = University of Neuchatel
2 LMJL - Laboratoire de Mathématiques Jean Leray

This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard-Helffer, Helffer-Persson-Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter h, we focus on the case when h is large. We hope to come back to the analysis when h is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with C ^(2,α) boundary (α > 0) as h large varies. MSC classification (2010): 35P99, 58J50, 58J37.

Mots clés: Courant-sharp; Robin eigenvalues; square; planar domains

Domaines: Mathématiques [math]; Théorie spectrale [math.SP]

K Gittins, Bernard Helffer. Courant-sharp Robin eigenvalues for the square and other planar domains. Portugaliae Mathematica, In press. ⟨hal-02369633⟩