= Bogomolny bound for the inverse-density Abelian Higgs model
{c}
{title2=$V\ge\pi N$}
Let $j_i=\operatorname{Im}(\bar\Phi D_i\Phi)$. The identity
$$
|D\Phi|^2=|(D_1+iD_2)\Phi|^2+Bh+\partial_1j_2-\partial_2j_1
$$
and <completing the square> give, when the current boundary integral vanishes,
$$
V=\frac12\int\left[
|(D_1+iD_2)\Phi|^2+
\frac{(B-\tfrac12h(1-h))^2}{h}\right]+\frac12\int B.
$$
For flux $2\pi N>0$, the lower bound is $\pi N$, saturated by the two first-order <Bogomolny equations>. Division by $h$ is interpreted in the finite-energy sense at its zeros.
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