Radu Ignat
Institut de Mathématiques de Toulouse, Université Paul Sabatier - Toulouse 3, France

We analyse vortex solutions to Ginzburg-Landau type systems depending on a small parameter $\epsilon>0$ in the unit ball $B^N$. The aim is to show the minimality of the symmetric solution $u:B^N\to \mathbb{R}^N$ corresponding to a vortex of degree one. We establish this minimality in dimension $N\geq 7$ for every $\epsilon>0$. In dimension $2\leq N\leq 6$, if $u:B^N\to \mathbb{R}^{N+1}$, we show a sharp dichotomy result between the minimality of the "non-escaping" vortex solution (i.e., confined in the space $\mathbb{R}^N\times \{0\}$) and the vortex solutions "escaping" in the $(N+1)$ direction according to the parameter $\epsilon$. Finally, we also discuss minimality of the vortex solution to the Ginzburg-Landau model for gradient fields. This is a series of works in collaboration with Luc Nguyen, Mircea Rus, Valeriy Slastikov and Arghir Zarnescu.