The Dirichlet-to-Neumann operator for divergence form problems

A. F.M. ter Elst*, G. Gordon, M. Waurick

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)
134 Downloads (Pure)

Abstract

We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one’s adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace spaces and are able to give meaning to the Dirichlet-to-Neumann operator of divergence form operators perturbed by a bounded potential in cases where the boundary of the underlying domain does not allow for a well-defined trace. Moreover, a representation of the Dirichlet-to-Neumann operator as a first-order system of partial differential operators is provided. Using this representation, we address convergence of the Dirichlet-to-Neumann operators in the case that the appropriate reciprocals of the leading coefficients converge in the weak operator topology. We also provide some extensions to the case where the bounded potential is not coercive and consider resolvent convergence.

Original languageEnglish
Pages (from-to)177-203
Number of pages27
JournalAnnali di Matematica Pura ed Applicata
Volume198
Issue number1
Early online date4 Jul 2018
DOIs
Publication statusPublished - 6 Feb 2019

Funding

Acknowledgements The authors wish to thank the referee for raising questions which improved the paper. The third named author expresses his gratitude for the wonderful atmosphere and hospitality extended to him during a two-month research visit at the University of Auckland. Part of this work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. Part of this work is supported by the EU Marie Curie IRSES program, Project AOS, No. 318910. Part of this work was carried out with financial support of the EPSRC grant EP/L018802/2: Mathematical foundations of metamaterials: homogenization, dissipation and operator theory.

Keywords

  • continuous dependence on the coefficients
  • Dirichlet-to-Neumann operator
  • resolvent convergence

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