CERAMATHS - DMATHS seminar : presentation by Damien Galant

The weekly seminar of the Department of Mathematics will host Damien Galant (UPHF and UMons, Belgium) on Thursday, December 15, for the following talk:

On the notion of "ground state" for the nonlinear Schrödinger equation on metric graphs

Summary:Initially, we will present the nonlinear Schrödinger equation (NLS) on metric graphs and its variational formulation. We will present the notion of "ground state", coming from the calculus of variations.
A "ground state" is a solution of a metric graph. A ground state is a solution obtained as a global minimizer of the functional under a mass constraint. In a second step, we will see that for non-compact graphs, we have to be careful when using the terminology "ground state". Indeed, minimizers under constraints of the action functional do not necessarily exist, because of the non-compactness. It is important to distinguish the notion of "ground state" from that of minimal action solution, a solution minimizing the functional among all the solutions of the equation. Four scenarios are a priori possible the ground states exist (and coincide with the minimal action solutions);
- the ground states do not exist but they are not the ground states do not exist but the minimum action solutions do;
- neither the ground states nor the minimum action solutions exist neither the ground states nor the minimal action solutions exist but the levels of the two associated minimization problems are equal;
neither the ground states nor the minimal action solutions exist but the levels of the two associated minimization problems are equal neither the ground states nor the minimal action solutions exist and the levels of the two associated minimization problems are different.
We will show that these four alternatives are equal. We will show that these four alternatives are possible in the context of metric graphs, by studying doubly constrained variational problems.
We will highlight the advantages of the metric graph framework over more classical frameworks such as that of unbounded Rⁿ openings in dimension N >= 2 for which it is currently unknown whether or not the four scenarios mentioned above take place.

This is a collaborative work with Colette De Coster (UPHF), Simone Dovetta (Politecnico di Torino) and Enrico Serra (Politecnico di Torino).

Retrouvez ici le programme complet du séminaire du département de mathématiques du CERAMATHS