Page 471 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 471
CA18232: VARIATIONAL METHODS AND EQUATIONS ON GRAPHS (MS-40)
Semigroups for flows on limits of graphs
Christian Budde, christian.budde@nwu.ac.za
North-West University, South Africa
Transport of goods is nowadays of extreme importance and indispensable considering what
mankind needs for daily life. Now imagine a start-up company shipping special goods all
over the world. Of course, the company starts with a small network of customers. However,
assuming the company grows and retains the already existing routes and customers, the ship
network grows and grows. It might come to the point in the development of the company, that
one actually lost the view on all specific routes but only knows how the network works since
it becomes too big. However, one still wants to know how the transport is going on the whole
network.
That a network is growing, through adding vertices and edges, means that one has a se-
quence of graphs and each graph is a subgraph of the subsequent graph in the sequence, de-
scribing the above mentioned situation of growing networks. We will describe the transition
from finite to infinite graphs by means of direct limits in a certain category. The approximation
of the transport process on the direct limit graph, is done by a version of the (first) Trotter–Kato
approximation theorem, which is originally due to T. Kato [2, Chapter IX, Thnm. 3.6] and H.F.
Trotter [3, Thm. 5.2 & 5.3] and modified by a version by K. Ito and F. Kappel [1, Thm. 2.1].
We will also present an extension the work of Ito and Kappel which is related to the well-known
second Trotter–Kato theorem.
References
[1] K. Ito and F. Kappel. The Trotter-Kato theorem and approximation of PDEs. Math. Comp.,
67(221):21–44, 1998.
[2] T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin-New York,
second edition, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132.
[3] H. F. Trotter. Approximation of semi-groups of operators. Pacific J. Math., 8:887–919,
1958.
Uniqueness and non-uniqueness of prescribed mass NLS ground states on
metric graphs
Simone Dovetta, simone.dovetta@imati.cnr.it
Centro IMATI - CNR, Italy
This talk addresses the problem of uniqueness of ground states of prescribed mass for the Non-
linear Schrödinger Energy with power nonlinearity on noncompact metric graphs with half–
lines. We first show that, up to an at most countable set of masses, all ground states at given
mass solve the same equation, that is the Lagrange multiplier appearing in the NLS equation is
constant on the set of ground states of mass µ. On the one hand, we apply this result to prove
uniqueness of ground states on two specific families of noncompact graphs. On the other hand,
we construct a graph that admits at least two ground states with the same mass having different
Lagrange multipliers. This shows that the result for Lagrange multipliers is sharp in general, in
the sense that one cannot get rid of the at most countable set of masses where it may fail without
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Semigroups for flows on limits of graphs
Christian Budde, christian.budde@nwu.ac.za
North-West University, South Africa
Transport of goods is nowadays of extreme importance and indispensable considering what
mankind needs for daily life. Now imagine a start-up company shipping special goods all
over the world. Of course, the company starts with a small network of customers. However,
assuming the company grows and retains the already existing routes and customers, the ship
network grows and grows. It might come to the point in the development of the company, that
one actually lost the view on all specific routes but only knows how the network works since
it becomes too big. However, one still wants to know how the transport is going on the whole
network.
That a network is growing, through adding vertices and edges, means that one has a se-
quence of graphs and each graph is a subgraph of the subsequent graph in the sequence, de-
scribing the above mentioned situation of growing networks. We will describe the transition
from finite to infinite graphs by means of direct limits in a certain category. The approximation
of the transport process on the direct limit graph, is done by a version of the (first) Trotter–Kato
approximation theorem, which is originally due to T. Kato [2, Chapter IX, Thnm. 3.6] and H.F.
Trotter [3, Thm. 5.2 & 5.3] and modified by a version by K. Ito and F. Kappel [1, Thm. 2.1].
We will also present an extension the work of Ito and Kappel which is related to the well-known
second Trotter–Kato theorem.
References
[1] K. Ito and F. Kappel. The Trotter-Kato theorem and approximation of PDEs. Math. Comp.,
67(221):21–44, 1998.
[2] T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin-New York,
second edition, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132.
[3] H. F. Trotter. Approximation of semi-groups of operators. Pacific J. Math., 8:887–919,
1958.
Uniqueness and non-uniqueness of prescribed mass NLS ground states on
metric graphs
Simone Dovetta, simone.dovetta@imati.cnr.it
Centro IMATI - CNR, Italy
This talk addresses the problem of uniqueness of ground states of prescribed mass for the Non-
linear Schrödinger Energy with power nonlinearity on noncompact metric graphs with half–
lines. We first show that, up to an at most countable set of masses, all ground states at given
mass solve the same equation, that is the Lagrange multiplier appearing in the NLS equation is
constant on the set of ground states of mass µ. On the one hand, we apply this result to prove
uniqueness of ground states on two specific families of noncompact graphs. On the other hand,
we construct a graph that admits at least two ground states with the same mass having different
Lagrange multipliers. This shows that the result for Lagrange multipliers is sharp in general, in
the sense that one cannot get rid of the at most countable set of masses where it may fail without
469
