Doubt about Sobolev space norm

Question

I consider the space $H^2(\mathbb{R}^3)$. I have a function and I have to verify that it belongs to this space. In the text I'm reading the author verifies that the function and its Laplacian are in $L^2$, not considering the firs order derivatives. So I remember that in $H^2$ I can consider the norm $$\Vert u\Vert=(\Vert u\Vert^2_{L^2}+\Vert\Delta u\Vert_{L^2}^2)^{\frac{1}{2}}$$ Is this norm equivalent to the usual norm $$\Vert u\Vert=\bigg(\sum_{|\alpha|\leq2}\Vert D^\alpha u\Vert^2_{L^2}\bigg)^{\frac{1}{2}}$$? If it is I understand the reasoning in the text.

Answer 1

The step you are missing is a so-called elliptic estimate for the laplacian, namely:

$$||u||_{H^2(\mathbb R^3)} \leq C(||\Delta u||_{L^2(\mathbb R^3)} + ||u||_{L^2(\mathbb R^3)}).$$

The proof is in Gilbarg and Trudinger's book for instance, and many other textbooks.