If $u \in H^2_0$ and $v \in H^1_0$

Question

If $u \in H^2_0$ and $v \in H^1_0$, I would like to show that $$\forall v \in H^1_0, \mbox{if}~a(u,v) = 0, \forall u \in H^2_0~\mbox{then}~v = 0,$$ where $a(u,v) = \langle Lu,v \rangle$ and $Lu = a(x) u^{(4)} + b(x) u''' + c(x) u'' + d(x) u' +e(x) u$. Also $a(x), b(x), c(x), d(x)$ and $e(x)$ are bounded nonzero functions on the interval $(0,1)$.