Page 475 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 475
8232: VARIATIONAL METHODS AND EQUATIONS ON GRAPHS (MS-40)
Stability and asymptotic properties of dissipative equations coupled with
ordinary differential equations
Serge Nicaise, serge.nicaise@uphf.fr
Université Polytechnique Hauts-de-France, France
In this talk, we will present some stability results of a system corresponding to the coupling
between a dissipative equation (set in an infinite dimensional space) and an ordinary differential
equation. Namely we consider U, P solution of the system
Ut = AU + MP, in H,
Pt = BP + N U, in X, (1)
U (0) = U0, P (0) = P0,
where A is the generator of a C0 semigroup in the Hilbert space H, B is a bounded operator
from another Hilbert space X, and M , N are supposed to be bounded operators. Many problems
from physics enter in this framework, let us mention dispersive medium models, generalized
telegraph equations, Volterra integro-differential equations, and cascades of ODE-hyperbolic
systems. The goal is to find sufficient (and necessary) conditions on the involved operators A,
B, M and N that garantee stability properties of system (1), i.e., strong stability, exponential
stability or polynomial one. We will illustrate our general results by an example of generalized
telegraph equations set on networks.
Self-adjoint extensions of infinite quantum graphs
Noema Nicolussi, noema.nicolussi@univie.ac.at
University of Vienna, Austria
In the last decades, quantum graphs (Laplacians on metric graphs) have become popular objects
of study and the analysis of spectral properties relies on the self-adjointness of the Laplacian.
Whereas on finite metric graphs the Kirchhoff Laplacian is always self-adjoint, much less is
known about the self-adjointness problem for graphs having infinitely many edges and vertices.
Intuitively the question is closely related to finding appropriate boundary notions for infinite
graphs.
In this talk we study the connection between self-adjoint extensions and the notion of graph
ends, a classical graph boundary introduced independently by Freudenthal and Halin. Our dis-
cussion includes a lower estimate on the deficiency indices of the minimal Kirchhoff Laplacian
and a geometric characterization of self-ajointness of the Gaffney Laplacian.
Based on joint work with Aleksey Kostenko (Ljubljana & Vienna) and Delio Mugnolo (Ha-
gen).
473
Stability and asymptotic properties of dissipative equations coupled with
ordinary differential equations
Serge Nicaise, serge.nicaise@uphf.fr
Université Polytechnique Hauts-de-France, France
In this talk, we will present some stability results of a system corresponding to the coupling
between a dissipative equation (set in an infinite dimensional space) and an ordinary differential
equation. Namely we consider U, P solution of the system
Ut = AU + MP, in H,
Pt = BP + N U, in X, (1)
U (0) = U0, P (0) = P0,
where A is the generator of a C0 semigroup in the Hilbert space H, B is a bounded operator
from another Hilbert space X, and M , N are supposed to be bounded operators. Many problems
from physics enter in this framework, let us mention dispersive medium models, generalized
telegraph equations, Volterra integro-differential equations, and cascades of ODE-hyperbolic
systems. The goal is to find sufficient (and necessary) conditions on the involved operators A,
B, M and N that garantee stability properties of system (1), i.e., strong stability, exponential
stability or polynomial one. We will illustrate our general results by an example of generalized
telegraph equations set on networks.
Self-adjoint extensions of infinite quantum graphs
Noema Nicolussi, noema.nicolussi@univie.ac.at
University of Vienna, Austria
In the last decades, quantum graphs (Laplacians on metric graphs) have become popular objects
of study and the analysis of spectral properties relies on the self-adjointness of the Laplacian.
Whereas on finite metric graphs the Kirchhoff Laplacian is always self-adjoint, much less is
known about the self-adjointness problem for graphs having infinitely many edges and vertices.
Intuitively the question is closely related to finding appropriate boundary notions for infinite
graphs.
In this talk we study the connection between self-adjoint extensions and the notion of graph
ends, a classical graph boundary introduced independently by Freudenthal and Halin. Our dis-
cussion includes a lower estimate on the deficiency indices of the minimal Kirchhoff Laplacian
and a geometric characterization of self-ajointness of the Gaffney Laplacian.
Based on joint work with Aleksey Kostenko (Ljubljana & Vienna) and Delio Mugnolo (Ha-
gen).
473
