Hi,


I am looking for the result:
$$\text{The norm} \quad \lVert \Delta u \rVert\_{L^2(S)} \quad \text{is equivalent to} \quad \lVert u \rVert\_{H^2(S)}$$
for scalar functions $u \in H^2(S)$, where $S$ is a compact hypersurface in $\mathbb{R}^n$, so in particular it's a compact manifold with no boundary. I know this result holds in the flat case (but have yet to prove that) for $H^2\_0$ functions so I am hoping it just carries through. 


Is this true? Can somebody give me a reference to this or show/tell me how to prove it?
I tried books by Aubin, Hebey and Michael E. Taylor but I did not find anything.f
Thanks.



Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold?
First, the correspondence 


$$u\mapsto\Vert \Delta u\Vert\_{L^2(S)} $$


is not a norm because if $u$ is constant, then $ \Delta u=0$. The correspondence


$$ u\mapsto \Vert u\Vert\_{L^2(S)}+ \Vert \Delta u\Vert\_{L^2(S)} $$


is a norm equivalent with $\Vert u\Vert\_{H^2(S)}$ because of the elliptic a priori estimates which state that if $L: C^\infty(S)\to C^\infty(S)$ is a 2nd order elliptic operator on $S$ with smooth coefficients, then there exists a constant $C>0$ such that


$$ \Vert u\Vert\_{H^2(S)}\leq C\Bigl(\; \Vert u\Vert\_{L^2(S)}+ \Vert L u\Vert\_{L^2(S)}\;\Bigr),\;\;\forall u\in H^2(S). $$


The Laplacian $\Delta: C^\infty(S)\to C^\infty(S)$ is such an operator.


