Authors

Marcus Waurick, Sven-Ake Wegner

Abstract

In this article we study port-Hamiltonian partial differential equations on certain one-dimensional manifolds. We classify those boundary conditions that give rise to contraction semigroups. As an application we study port-Hamiltonian operators on networks whose edges can have finite or infinite length. In particular, we discuss possibly infinite networks in which the edge lengths can accumulate zero and port-Hamiltonian operators with Hamiltonians that neither are bounded nor bounded away from zero. We achieve this, by first providing a new description for maximal dissipative extensions of skew-symmetric operators. The main technical tool used for this is the notion of boundary systems. The latter generalizes the classical notion of boundary triple(t)s and allows to treat skew-symmetric operators with unequal deficiency indices. In order to deal with fairly general variable coefficients, we develop a theory of possibly unbounded, non-negative, injective weights on an abstract Hilbert space.

Keywords

C 0 -semigroup; Port-Hamiltonian PDEs with singular weights; Maximal dissipative operators; Differential equations on infinite networks; Boundary triple(t); Quantum Graphs with vanishing edge lengths

Citation

  • Journal: Journal of Differential Equations
  • Year: 2020
  • Volume: 269
  • Issue: 9
  • Pages: 6830–6874
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.jde.2020.05.014

BibTeX

@article{Waurick_2020,
  title={{Dissipative extensions and port-Hamiltonian operators on networks}},
  volume={269},
  ISSN={0022-0396},
  DOI={10.1016/j.jde.2020.05.014},
  number={9},
  journal={Journal of Differential Equations},
  publisher={Elsevier BV},
  author={Waurick, Marcus and Wegner, Sven-Ake},
  year={2020},
  pages={6830--6874}
}

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References