The problem can be formulated as following
Let $L$ be a second order differential operator $L = > a^{ij}D_{ij}+b^iD_i+c$ where Einstein summation is imposed. And the underlying field for function spaces is $\mathbb{R}^d$.
Assume $\exists \lambda, N_0\in (0,\infty) $ such that for $\forall > u\in C^2_0$ and $t\in[0,1]$, we have $\|u\|_{W^{2,2}}\leq N_0 \|L_t u\|_2,$ where $L_t = (1-t)(\lambda-\Delta)+tL$ and $W^{2,2}$ is Sobolev space.
I want to extend the inequality to $u\in W^{2,2}.$ It is easy to show that $Lt$ is bounded.
And since $C^2_0$ dense in $W^{2,2}$, take $u^n \in C^2_0$ s.t. $u^n \rightarrow u$ ( in the sense of $W^{2,2}$).
we have $\|u\|_{W^{2,2}} \leq \|u-u^n\|_{W^{2,2}}+\|u^n\|_{W^{2,2}} \leq |u-u^n\|_{W^{2,2}} + N_0\|L_t u^n\|_2.$
And here is my question:
How to show $\lim_{n\rightarrow \infty}\|L_t u^n\|_2 \leq \|\lim_{n\rightarrow \infty}L_t u^n\|_2 $?
Reverse Fatou lemma seems to be a good option, but I don't know how to show it is valid in this case.