Consider a differential operator $L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + \sum_{i,j}\|u_{i,j}\|_2^2 =\|\Delta u-\lambda u\|^2_2\leq \frac{1}{\mu^2}\|Lu\|_2^2$
where $u_{i} = \frac{\partial u}{\partial x_i}$, $u_{i,j} = \frac{\partial^2 u}{\partial x_i \partial x_j}$, for $i=1,2;\,j=1,2$, and constant $\mu>0$.
My question is if it is possible to show the following inequality: $\|u\|_{W^{2,2}}\leq N_0 \|L_t u\|_2$
where$\|u\|_{W^{2,2}}$ is the Sobolev norm and constant $N_0$ is independent of $u$.
The question is originally from Krylov's book, ex 9 on page 17. I'm trying apply theorem 4(method of continuity) on page 15. Should it work, or I'm wrong from the very beginning?
Thank you in advance.