# Elliptic regularity on compact manifold without boundary

Let $$(M,g)$$ be a Riemannian compact manifold without boundary, and $$\Delta$$ is the Laplace-Beltrami operator on $$M$$. Is there any result on the elliptic regularity like this:

For any $$u\in H^1(M)$$, and $$f\in L^2(M)$$ such that $$\Delta u = f$$ (in the sens of distributions), Then $$u \in H^2(M)$$. If there is a nice reference for such regularity result It would be good.

• I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds. – Neal Dec 7 at 2:26
• I found Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. In Taylor I. But I don't know if this implies the desired résult. He assume that $u|_{\partial M} =0$, in my case there is no boundary. – S. Cho Dec 7 at 19:35

## 3 Answers

This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $$\Omega \subset \mathbb{R}^n$$, there exists $$C>0$$ such that for any function (or even just a distribution) $$u$$ compactly supported in $$\Omega$$, $$\tag{*} |u|_{H^2} \le C|\Delta_0 u|_{L^2},$$ where $$\Delta_0$$ is the standard flat Laplacian.

To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $$u$$ compactly supported on a neighborhood of each point $$p \in M$$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $$p$$, then you can assume that the Laplace-Beltrami operator is of the form $$\Delta u = (\delta^{ij} + a^{ij}(x))\partial^2 + b^k\partial_ku$$ where $$|a^{ij}|, |b_k| < \epsilon << 1$$. Therefore, if $$\Delta_g u = f$$, then $$\Delta_0u = -a_{ij}\partial^2_{ij}u - b^k\partial_ku + f$$ Therefore, by $$(*)$$ $$|u|_{H^2} \le C(\epsilon |u|_{H^2} + |f|_{L^2}).$$ If the neighborhood is sufficiently small, then $$C\epsilon < 1$$ and therefore, $$|u|_{H^2} \le C|f|_{L^2}.$$

• @Yang Thank you ! Can you recommend a good reference for such proof ? – S. Cho Dec 7 at 10:32
• Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that. – Deane Yang Dec 7 at 16:45
• There is a similar result in Taylor's book when the $u\in H^1_0(M)$. Is this implies the result for my case ? – S. Cho Dec 7 at 17:22
• I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book. – Deane Yang Dec 7 at 17:39
• It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. – S. Cho Dec 7 at 19:31

This result is true. This is Theorem 6.30 in:

F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.

While there are many books that deal with elliptic regularity on manifolds, Warner's book seems most elementary and oriented towards those who do not know much about analysis, but are familiar with geometry of manifolds.

• I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces. – Mike Miller Dec 7 at 2:48
• @Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book. – S. Cho Dec 7 at 10:14
• @S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references. – Piotr Hajlasz Dec 7 at 14:35
• @PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery. – Deane Yang Dec 7 at 16:48

A more general theorem with a proof using pseudodifferential operators is Theorem 7.2 in Shubin's book (Pseudodifferential operators and Spectral Theory). In your case the operator is second order and elliptic, so $$m=m_0 = 2$$ and $$\rho=1, \delta=0$$.