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Description

We minimize the standard Ginzburg-Landau functional $E_\epsilon$ over maps $u$ defined on the unit disk with values into ${\mathbb R}^3$ such that at the boundary $u$ takes values into the horizontal equator with $u=(cos k\theta, sin k\theta, 0)$ for the nonzero winding number k. It is known that the limiting problem as $\epsilon$ tends to $0$ (i.e., the harmonic map problem) has exactly two minimizers that take values either on the upper or lower hemisphere and these minimizers are symmetric. We will show that for small $\epsilon>0$, the Ginzburg-Landau functional has also exactly two minimizers that are radially symmetric. This is a joint work with Luc Nguyen, Valeriy Slastikov and Arghir Zarnescu.