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Mathematics > Spectral Theory

arXiv:1812.09344 (math)
[Submitted on 21 Dec 2018 (v1), last revised 8 Feb 2019 (this version, v2)]

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

Authors:Katie Gittins, Bernard Helffer
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Abstract: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.
Subjects: Spectral Theory (math.SP)
MSC classes: 35P99, 58J50, 58J37
Cite as: arXiv:1812.09344 [math.SP]
  (or arXiv:1812.09344v2 [math.SP] for this version)
  https://doi.org/10.48550/arXiv.1812.09344

Submission history

From: Katie Gittins [view email]
[v1] Fri, 21 Dec 2018 19:21:37 UTC (185 KB)
[v2] Fri, 8 Feb 2019 13:02:41 UTC (176 KB)
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