Cursos
Concentration and blow-up via asymptotic gluing
Ponente: Marco Badran (ETH Zürich)Fecha: martes 07 de mayo de 2024 - miércoles 08 de mayo de 2024 - 11:00-13:00Lugar: Aula 112 (Euler), ICMAT
Resumen:
It often happens that solutions to nonlinear PDEs exhibit a very distinct feature: in certain regions of their domain, they closely resemble rescaled versions of an entire solution to the same equation. These solutions concentrate around a specific set of points or submanifolds, named the concentration set. So the question is natural: given an equation and a subset of the domain, can we find a solution that concentrate exactly on this subset? As one expects the answer is, in general, negative. In most cases, the concentration set necessitates specific criticality conditions and cannot be arbitrarily chosen!
In this short-course I will explain the basic ideas behind some gluing techniques based on the Lyapunov-Schmidt reduction method. We will see that in many instances is actually possible to reduce the problem of finding a solution with a prescribed concentration set to a problem for the concentration set itself. These techniques give rise to new solutions to an incredible variety of equations, ranging from phase transition to fluid dynamics, from prescribed curvature to superconductivity and superfluids. I will present some examples and describe the technique in a functional framework.
