CERAMATHS - DMATHS seminar: presentation by Teresa Dos Santos

The CÉRAMATHS mathematics department seminar will welcome, exceptionally at 3:15 pm, Teresa Oitaben Santos (CÉRAMATHS, UPHF), Thursday March 12, 2026, for the following talk:

Standardized solutions: an abstract formulation

In this talk, we will discuss a new approach to studying normalized solutions (i.e., whose L² norm is fixed) of an abstract superlinear Schrödinger equation.
The classical approach consists in transforming the problem of the existence of a solution into the search for a critical point of a functional, called energy, restricted to the set of functions of fixed L² norm. We thus obtain a solution to an equation in which a linear term with a Lagrange multiplier appears. The lack of explicit knowledge of this Lagrange multiplier leads to difficulties in certain cases.
We will show that it is possible to adopt a different point of view: instead of imposing the mass constraint from the outset, we can study the mass of the solutions of the problem obtained by fixing the Lagrange multiplier.
Even in this new framework, the mass of the solutions of the problem obtained by fixing the Lagrange multiplier can be studied. Even in this new framework, the analysis remains variational in nature. We are looking for critical points of a new functional, called action. As this new functional is not bounded inferiorly over all space, we restrict the study to a natural constraint associated with the functional (Nehari variety), over which it becomes legitimate to minimize the functional. Moreover, this set has the property of containing all the non-trivial critical points of this functional, which makes it possible to formulate the search for solutions as a well-posed minimization problem.
In a second step, we will specify the conditions ensuring the validity of this abstract framework, and illustrate the approach through examples.