Gårding's inequality ($B[v,v]\ge a\Vert v{\Vert }_{H^1}^2-b\Vert v{\Vert }_2^2$)

For a uniformly elliptic second-order Dirichlet form with bounded coefficients and lower-order terms, there are $a>0$ and $b\ge 0$ giving the displayed lower bound. 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.