From Steklov to Neumann and Beyond, via Robin: The SzegÅ' Way

Pedro Freitas, Richard S. Laugesen

Research output: Contribution to journalArticlepeer-review


The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form, and lies between and. Corollaries include SzegÅ"s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock's inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter. The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

Original languageEnglish (US)
Pages (from-to)1024-1043
Number of pages20
JournalCanadian Journal of Mathematics
Issue number4
StatePublished - Aug 1 2020


  • AMS subject classification: 35P15 30C70

ASJC Scopus subject areas

  • Mathematics(all)


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