### 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.

Language | English |
---|---|

Pages | 177-203 |

Number of pages | 27 |

Journal | Annali di Matematica Pura ed Applicata |

Volume | 198 |

Issue number | 1 |

Early online date | 4 Jul 2018 |

DOIs | |

Publication status | Published - 6 Feb 2019 |

### Fingerprint

### Keywords

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

### Cite this

*198*(1), 177-203. https://doi.org/10.1007/s10231-018-0768-2

}

**The Dirichlet-to-Neumann operator for divergence form problems.** / ter Elst, A. F.M.; Gordon, G.; Waurick, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Dirichlet-to-Neumann operator for divergence form problems

AU - ter Elst, A. F.M.

AU - Gordon, G.

AU - Waurick, M.

PY - 2019/2/6

Y1 - 2019/2/6

N2 - 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.

AB - 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.

KW - continuous dependence on the coefficients

KW - Dirichlet-to-Neumann operator

KW - resolvent convergence

UR - http://www.scopus.com/inward/record.url?scp=85049552232&partnerID=8YFLogxK

UR - https://strathprints.strath.ac.uk/61320/

UR - https://arxiv.org/abs/1707.05734

U2 - 10.1007/s10231-018-0768-2

DO - 10.1007/s10231-018-0768-2

M3 - Article

VL - 198

SP - 177

EP - 203

IS - 1

ER -