This question has been asked here showing $\lVert \Delta u \rVert_{L^2(U)} + \lVert u \rVert_{L^2(U)}$ is equivalent to $\|u\|_{H^2(U)}$ norm for $H^2(U)$ space assuming the domain $U$ is sufficient smooth. However I came across some problem when following the hint below the post, I do as follows:
Consider $u \in H^2(U)$ it satisfies the equation :
$$\Delta u = f \\ u = g\ \text{ on }\ \partial U $$
By the elliptic reguarity we have $$\|u\|_{H^2} \le C(\|u\|_{L^2 } + \|f\|_{L^2} + \|g\|_{H^2(U)})$$
The problem here differs from the problem for equivalent norm in $H = H^1_0 \cap H^2$ is that this equation above does not has homogenuous boundary condition, therefore the regularity result has $\|g\|_{H^2(U)}$ in it correct.I have no idea how to bound this term.