Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold? 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.
 A: 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.
A: I wonder if what you're looking for is just the following simple identity. Say $u$ is either a periodic function, or a function with compact support in $\mathbb R^d$, then we integrate by parts to obtain
$$
\int |\Delta u|^2 = \int \partial_{ii} u \partial_{jj} u = - \int \partial_i u \partial_{ijj} u = \int \partial_{ij} u \partial_{ij} u = \int |D^2 u|^2$$
If you give the right definitions to $\Delta u$ and $D^2 u$ on any compact Riemannian manifold $S$, a similar computation should go through.
