An inequality for the Steklov spectral zeta function of a planar domain

Alexandre Jollivet, Vladimir Sharafutdinov

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2 Citations (Scopus)


We consider the zeta function Ω for the Dirichlet-to-Neumann operator of a simply connected planar domain Ωbounded by a smooth closed curve. We prove that, for a fixed real s satisfying jsj > 1 and fixed length L.@ Ω/ of the boundary curve, the zeta function Ω.s/ reaches its unique minimum when Ωis a disk. This result is obtained by studying the difference Ω(s)-2L.@ Ω/ 2 π R.s/,where R stands for the classicalRiemann zeta function. The difference turns out to be non-negative for real s satisfying jsj > 1. We prove some growth properties of the difference as s →±∞ Two analogs of these results are also provided.

Original languageEnglish
Pages (from-to)271-296
Number of pages26
JournalJournal of Spectral Theory
Issue number1
Publication statusPublished - 1 Jan 2018


  • Dirichlet-to-Neumann operator
  • Inverse spectral problem
  • Steklov spectrum
  • Zeta function
  • inverse spectral problem
  • zeta function


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