I'm studying Sobolev spaces using Evans' PDE book. I can't figure out this simple fact. Let $L$ be an operator in this form: $$Lu= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}.$$

I can't understand why $L \in H^{-1}$, the dual space of $H_0^1$. I'm struggling because to me it would require that $u$ must have 2 derivatives so $u \in H^2$ to be well defined. Am I missing something? Thanks.