Application of method of continuity in partial differential equations

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 , 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.

2022-07-25 20:41:11
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