The Dirichlet-to-Neumann operator for divergence form problems

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

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5 Citations (Scopus)
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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

Keywords

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

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