= Magnetic-energy lower bound from helicity
{title2=$E_B\geq |H_M|/(2\mu_0 C_D)$}
In a bounded regular domain, fix the global extension or circulation convention for the <magnetic vector potential>. A bounded inverse of <curl> on tangent <solenoidal vector fields> gives $\|\mathbf A\|_2\leq C_D\|\mathbf B\|_2$. The <Cauchy-Schwarz inequality> then gives $|H_M|\leq C_D\|\mathbf B\|_2^2=2\mu_0 C_DE_B$. One construction extends the tangent field by zero outside the domain and uses the whole-space <Biot-Savart law>; its kernel of size $|\mathbf x-\mathbf y|^{-2}$ is integrable over a bounded region, giving the required $L^2$ bound. Nonzero conserved <magnetic helicity> therefore prevents complete decay of <magnetic energy>.
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