= Solution
Finite energy requires $|\phi|\to1$ and $D\phi\to0$ on the circle at spatial infinity. Writing $\phi\sim e^{i\chi}$ there gives $A\sim d\chi$. The <vortex number> is the <winding number>
$$
N=\frac1{2\pi}\oint_{S_\infty^1}d\chi\in\mathbb Z.
$$
By <Stokes theorem>, the <magnetic flux> is quantized:
$$
\boxed{\int_{\mathbb R^2}B\,dx^1\wedge dx^2
=\oint_{S_\infty^1}A=2\pi N}.
$$
For the rotationally symmetric <Abelian Higgs vortex> ansatz, $A=f(r)d\theta$ and $\phi=h(r)e^{ik\theta}$ give
$$
B=\frac{f'}r,
\qquad
|D\phi|^2=(h')^2+\frac{(k-f)^2h^2}{r^2}.
$$
After the angular integration, the <Abelian Higgs model> energy becomes
$$
\mathcal E(f,h)=\pi\int_0^\infty
\left[
\frac{(f')^2}{r}+r(h')^2
+\frac{(k-f)^2h^2}{r}
+\frac r4(1-h^2)^2
\right]dr.
$$
<Completing the square> in the two pairs of terms gives
$$
\begin{aligned}
\frac{\mathcal E}{\pi}
={}&\int_0^\infty\left[
r\left(h'-\frac{k-f}{r}h\right)^2
+\frac1r\left(f'-\frac r2(1-h^2)\right)^2
\right]dr\\
&-\left[(k-f)(1-h^2)\right]_{0}^{\infty}.
\end{aligned}
$$
Regularity at the origin and approach to the vacuum at infinity require
$$
\boxed{f(0)=0,\quad h(0)=0,\qquad
f(\infty)=k,\quad h(\infty)=1}.
$$
More precisely, $h(r)=O(r^k)$ and $f(r)=O(r^2)$ near the origin. The boundary term is $k$, so
$$
\boxed{\mathcal E\geq k\pi}.
$$
The bound is saturated exactly when both squares vanish, giving the radial <Bogomolny vortex equations>
$$
\boxed{
h'=\frac{k-f}{r}h,
\qquad
f'=\frac r2(1-h^2)}.
$$
The asymptotic value $f(\infty)=k$ also makes the flux $2\pi k$, so the ansatz has vortex number $N=k$.
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