Program

These lectures will provide an overview on the theory of normalized ground states for nonlinear Schrödinger equations on metric graphs, defined as global minimizers of the energy among functions with prescribed mass (i.e. the L^2 norm). In particular, we will focus on existence and nonexistence results for ground states on general graphs. In recent years, this problem has been proved to be highly sensitive to the specific structure of the graph under exam, exhibiting a rich phenomenology. In these lectures, we will first illustrate a general existence argument for ground states, rephrasing in the context of metric graphs the typical framework of concentration-compactness. This general argument will be then applied to different families of graphs (graphs with half-lines, periodic graphs,...) to unravel how topological and metric properties affect the problem by creating obstructions to the existence of ground states. 

In population dynamics, a pivotal class of models for the dispersal of a species in a heterogeneous environment is based on logistic reaction-diffusion equations. To describe the environment divided into different zones, a weight that changes sign is introduced, so that the positive weight zones correspond to the zones favoring survival. When the equation is associated with homogeneous boundary conditions, the problem presents a threshold for the persistence of the population, encoded by a principal eigenvalue of the associated linearized equation. The problem of minimization of this eigenvalue therefore appears naturally in this context.

During the course, we will consider various models in this context and the derivation of the related optimization problems, illustrating the main open questions and possible lines of research.

In particular, we will focus on the case when the minimization is set on a suitable class of weights, the minimum is reached by a piecewise constant (bang-bang) control, and thus the problem translates in a shape optimization/free boundary one. The qualitative properties of the optimal set are

completely understood only in the one-dimensional case while many questions are open in higher dimensions, and can be attacked, for instance, by singular perturbation analysis and quantitative estimates.