Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains

Alexandre Jollivet, Vladimir Sharafutdinov

Research output: Contribution to journalArticlepeer-review

Abstract

The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function a on the unit circle from the spectrum of the operator aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Zm(a)=Tr[(aΛ)2m−(aD)2m] for every smooth function a. In the case of a positive a, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for Zm(a) in the case of a real function a. On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the C-topology. We also describe all real functions a satisfying Zm(a)=0.

Original languageEnglish
Pages (from-to)1712-1755
Number of pages44
JournalJournal of Functional Analysis
Volume275
Issue number7
DOIs
Publication statusPublished - 1 Oct 2018

Keywords

  • Dirichlet-to-Neumann operator
  • Inverse spectral problem
  • Steklov spectrum
  • Zeta function
  • SETS
  • NEUMANN OPERATOR

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