We consider a port-Hamiltonian system on an open spatial domain Ω ⊆ R n with bounded Lipschitz boundary. We show that there is a boundary triple associated to this system. Hence, we can characterize all boundary conditions that provide unique solutions that are non-increasing in the Hamiltonian. As a by-product we develop the theory of quasi Gelfand triples. Adding "natural" boundary controls and boundary observations yields scattering/impedance passive boundary control systems. This framework will be applied to the wave equation, Maxwell's equations and Mindlin plate model. Probably, there are even more applications.
For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any $$\mathsf {L}_{}^{2}$$
L
2
-bounded sequence of vector fields with $$\mathsf {L}_{}^{2}$$
L
2
-bounded rotations and $$\mathsf {L}_{}^{2}$$
L
2
-bounded divergences as well as $$\mathsf {L}_{}^{2}$$
L
2
-bounded tangential traces on one part of the boundary and $$\mathsf {L}_{}^{2}$$
L
2
-bounded normal traces on the other part of the boundary, contains a strongly $$\mathsf {L}_{}^{2}$$
L
2
-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.
In the present note a spectral theorem for a finite tuple of pairwise commuting, self-adjoint and definitizable bounded linear operators $$A_1,\ldots ,A_n$$
A
1
,
…
,
A
n
on a Krein space is derived by developing a functional calculus $$\phi \mapsto \phi (A_1,\ldots ,A_n)$$
ϕ
↦
ϕ
(
A
1
,
…
,
A
n
)
which is the proper analogue of $$\phi \mapsto \int \phi \, dE$$
ϕ
↦
∫
ϕ
d
E
in the Hilbert space situation with the common spectral measure E for a finite tuple of pairwise commuting, self-adjoint bounded linear operators.
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