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I'm working on my master thesis and need to handle some spectral theory of the Laplace operator on compact Riemannian manifolds and especially on the sphere. While investigating essential self-adjointness I stumbled on the following problem.

*Problem*$\quad$ In a compact Riemannian manifold $M$ let $$\Delta=\operatorname{div}\operatorname{grad}$$ and let $f\in L^2(M)$ be such that $(f, u-\Delta u)=0$ for every $u \in C^{\infty}(M)$. Prove that $f=0$.

I believe that the claim is true, because the condition $(f, u-\Delta u)=0$ means exactly that $f$ is a distributional solution of the elliptic equation $-\Delta f + f=0$, and so I expect it to be a $H^2_{\text{loc}}$ function (see Theorem 2.1 of Berezin - Shubin's book). Since $M$ is compact this must imply that $f\in H^1(M)$ so that integrating by parts we get $\lVert f \rVert_{H^1}^2=(f, f)+(\operatorname{grad}f, \operatorname{grad}f)=0$.

Unfortunately Theorem 2.1 above is set in an open subset of the Euclidean space and I don't know if it is applicable verbatim in a Riemannian manifold. Can you point me to some reference on this?

Thank you.

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I don't know the answer to this but would first try to look in Chavel's book 'Eigenvalues in Riemannian Geometry'. – user20266 Jun 28 '12 at 18:07
up vote 4 down vote accepted

Yes. Write your equation in local coordinates and then use the usual elliptic theory there.

Here is an example. The equation $$\Delta_g u = h$$ in local coordinates is $$g^{ij}\frac{\partial^2u}{\partial x^i\partial x^j} - \frac{1}{\sqrt g} \frac{\partial}{\partial x^i}(\sqrt g g^{ij})\frac{\partial u}{\partial x^j} = h$$

Notice that the operator on the LHS is still an elliptic operator on $\mathbb R^n$ in the given local coordinates, due to the fact that the Riemannian metric is positive definite. Therefore all of the standard elliptic regularity theorems you know for operators on $\mathbb R^n$ still apply.

There isn't really a standard reference for this, although it probably appears as a remark buried in most PDE books.

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