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Journal Article

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


Gittins,  Katie
Max Planck Institute for Mathematics, Max Planck Society;

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Gittins, K., & Helffer, B. (2019). Courant-sharp Robin eigenvalues for the square and other planar domains. Portugaliae Mathematica, 76(1), 57-100. doi:10.4171/PM/2027.

Cite as: https://hdl.handle.net/21.11116/0000-0004-D924-F
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 $C^{2,\alpha}$ boundary ($\alpha >0$) as $h$ large varies.