SEMINARIO DOBLE DE ECUACIONES EN DERIVADAS PARCIALES - Global bifurcation for corotating vortex pairs // Null-Lagrangians and callibrations for nonlocal elliptic functionals

Speaker:  Claudia García, UAM // Juan Carlos Felipe Navarro, U. Helsinki ()
Date:  Friday, 07 October 2022 - 10:00 - 12:30
Place:  Aula 520, Módulo 17, Departamento de Matemáticas, UAM


10:00. "Global bifurcation for corotating vortex pairs", Claudia García (UAM)

Abstract: The existence of a local curve of corotating vortex pairs was proven by Hmidi and Mateu via a desingularization of a pair of point vortices. In this talk, we will present a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of trivial solutions. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland, to allow for the singularity at the bifurcation point. This is a collaboration with Susanna V. Haziot.


11:00. Coffee break


11:30. "Null-Lagrangians and callibrations for nonlocal elliptic functionals", Juan Carlos Felipe Navarro (U. Helsinki)

Abstract: This talk will be devoted to introduce a null-Lagrangian and a calibration for nonlocal elliptic functionals in the presence of an extremal field.
First, I will review the classical Weierstrass theory of extremal fields from the Calculus of Variations. Next, I will explain how to extend it to the nonlocal setting, where our model functional is the one associated to the fractional Laplacian. Finally, I will show as applications that monotone solutions are minimizers and that minimizers are viscosity solutions.
This is a joint work with Xavier Cabré (ICREA-UPC-CRM) and Iñigo U. Erneta (UPC-BGSMath)