= Gårding's inequality
{c}
{title2=$B[v,v]\ge a\|v\|_{H^1}^2-b\|v\|_2^2$}
For a uniformly elliptic second-order <Dirichlet boundary condition>[Dirichlet] form with <bounded> coefficients and lower-order terms, there are $a>0$ and $b\ge0$ giving the displayed lower bound. <Cauchy-Schwarz inequality>[Cauchy-Schwarz] and Young's inequality absorb the drift term into part of the principal <gradient> energy plus an $L^2$ correction. Adding a sufficiently large nonnegative multiple of the $L^2$ <inner product> therefore makes the form coercive, permitting Lax-Milgram. The shift restores coercivity even when the unshifted form is not positive.
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