Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it.

For example, for $L$ a linear elliptic operator, (eg. $L = \Delta$), I want to know that $$\lVert u \rVert_{H^2(M)} \leq C(\lVert Lu \rVert_{L^2(M)} + \lVert u \rVert_{L^2(M)})$$ holds where $M$ is a compact (boundaryless) Riemannian manifold.

I am interested in general nth order elliptic estimates, but a source for the above would be good as well. Thanks.

share|cite|improve this question
It doesn't suffice to use equivalent estimates on bounded domains of $\mathbb{R}^n$, and then use a partition of unity to extend it to the manifold? – Ray Yang Feb 25 '13 at 2:33
@RayYang Change of variables introduces extra first-order terms (to be controlled by $Lu$), which looks like a bit of extra work. It should be written somewhere, I'm sure. – user53153 Feb 25 '13 at 3:18
@RayYang Also I am not that familiar with manifolds and PDEs and have never seen such a POI trick to extend a result. – george.s Feb 26 '13 at 9:41

3 Answers 3

up vote 2 down vote accepted

Lawson, Michelsohn: Spin Geometry, Princeton University Press, 1989.

The needed norm estimate is Theorem 5.2(iii) in Chapter III, §5.

share|cite|improve this answer
Thanks. I didn't think a book called Spin Geometry would have such a result! – george.s Feb 27 '13 at 18:53

How about this:

Try Proposition 9 on page 43. Anybody have any comments on this source?

share|cite|improve this answer

I don't know if it is still relevant, but in Gilkeys "Invariance Theory, the Heat equation and the Atiyah-Singer Index Theorem", the results are established in full generality (for any Exponent and any elliptic (pseudo)differential opertator).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.