JournalsjstVol. 12, No. 1pp. 195–225

The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

  • Alexandre Girouard

    Université Laval, Québec, Canada
  • Mikhail Karpukhin

    California Institute of Technology – Caltech, Pasadena, USA
  • Michael Levitin

    University of Reading, UK
  • Iosif Polterovich

    Université de Montréal, Canada
The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript cover
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Abstract

How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.

Cite this article

Alexandre Girouard, Mikhail Karpukhin, Michael Levitin, Iosif Polterovich, The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript. J. Spectr. Theory 12 (2022), no. 1, pp. 195–225

DOI 10.4171/JST/399