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 , we focus on the case where is large. We consider the case where 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 piecewise boundary as large varies.
Cite this article
Katie Gittins, Bernard Helffer, Courant-sharp Robin eigenvalues for the square and other planar domains. Port. Math. 76 (2019), no. 1, pp. 57–100DOI 10.4171/PM/2027