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
}
The Dirichlet-to-Neumann operator for divergence form problems. / ter Elst, A. F.M.; Gordon, G.; Waurick, M.
Vol. 198, No. 1, 06.02.2019, p. 177-203.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 -