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Séminaire de Physique Mathématique - EDP
Responsables : Jean-Baptiste Burie, Franck Sueur
Le 7 mai 2024
à 11:00
Séminaire de Physique Mathématique - EDP
Zoom: https://cnrs.zoom.us/j/91463344125?pwd=WVBtNjUzSnBqWXI3SDlYZTN5akc2dz09
Radu Ignat Institut de Mathématiques de Toulouse
Minimality of the vortex solution for Ginzburg-Landau systems
We consider the standard Ginzburg-Landau system for N-dimensional maps defined in the unit ball for some parameter eps>0. For a boundary data corresponding to a vortex of topological degree one, the aim is to prove the (radial) symmetry of the ground state of the system. We show this conjecture in any dimension N≥7 and for every eps>0, and then, we also prove it in dimension N=4,5,6 provided that the admissible maps are curl-free. This is part of joint works with L. Nguyen, M. Rus, V. Slastikov and A. Zarnescu.
Le 14 mai 2024
à 11:00
Séminaire de Physique Mathématique - EDP
Salle de conférences
David Krejcirik Czech Technical University in Prague\,
Is the optimal rectangle a square?
We give a light talk on optimality of shapes in geometry and physics. First, we recollect classical geometric results that the disk has the largest area (respectively, the smallest perimeter) among all domains of a given perimeter (respectively, area). Second, we recall that the circular drum has the lowest fundamental tone among all drums of a given area or perimeter and reinterpret the result in a quantum-mechanical language of nanostructures. In parallel, we discuss the analogous optimality of square among all rectangles in geometry and physics. As the main body of the talk, we present our recent attempts to prove the same spectral-geometric properties in relativistic quantum mechanics, where the mathematical model is a matrix-differential (Dirac) operator with complex (infinite-mass) boundary conditions. It is frustrating that such an illusively simple and expected result remains unproved and apparently out of the reach of current mathematical tools.
Les séminaires depuis 2013