Author
Y. Arlinskiĭ
Bio: Y. Arlinskiĭ is an academic researcher. The author has contributed to research in topics: Boundary value problem & Fourier integral operator. The author has an hindex of 1, co-authored 1 publications receiving 22 citations.
Papers
More filters
••
01 Oct 2012
22 citations
Cited by
More filters
••
TL;DR: In this article, a purely functional analytic framework for elliptic boundary value problems in a variational form is introduced, and a theory relating resolvents and spectra of these operators is developed.
Abstract: We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet-to-Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non-smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.
45 citations
••
TL;DR: In this article, the linear wave equation on an n-dimensional spatial domain was studied and it was shown that there is a boundary triplet associated to the undamped wave equation.
Abstract: In this paper, we study the linear wave equation on an n-dimensional spatial domain. We show that there is a boundary triplet associated to the undamped wave equation. This enables us to characterise all boundary conditions for which the undamped wave equation possesses a unique solution non-increasing in the energy. Furthermore, we add boundary inputs and outputs to the system, thus turning it into an impedance conservative boundary control system.
23 citations
••
TL;DR: In this paper, the spectral properties of non-self-adjoint extensions of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions.
21 citations
••
TL;DR: In this article, a self-contained and streamlined exposition of a generation theorem for C0-semigroups based on the method of boundary triplets is given, and applied to port-Hamiltonian systems where they discuss recent results appearing in stability and control theory.
Abstract: We give a self-contained and streamlined exposition of a generation theorem for C0-semigroups based on the method of boundary triplets. We apply this theorem to port-Hamiltonian systems where we discuss recent results appearing in stability and control theory. We give detailed proofs and require only a basic knowledge of operator and semigroup theory.
10 citations
••
TL;DR: In this paper, a method of determining the proper dissipative extensions of a dual pair of operators on a Hilbert space is presented. But the stability of the numerical range of the different extensions is not investigated.
Abstract: Let A and $${(-\widetilde{A})}$$
be dissipative operators on a Hilbert space $${\mathcal{H}}$$
and let $${(A,\widetilde{A})}$$
form a dual pair, i.e. $${A \subset \widetilde{A}^*}$$
, resp. $${\widetilde{A} \subset A^*}$$
. We present a method of determining the proper dissipative extensions $${\widehat{A}}$$
of this dual pair, i.e. $${A\subset \widehat{A} \subset\widetilde{A}^*}$$
provided that $${\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})}$$
is dense in $${\mathcal{H}}$$
. Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.
9 citations