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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.

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    $\begingroup$ I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds. $\endgroup$
    – Neal
    Dec 7, 2018 at 2:26
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    $\begingroup$ 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. $\endgroup$
    – S. Maths
    Dec 7, 2018 at 19:35

4 Answers 4

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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, $$ |u|_{H^2} \le C|f|_{L^2}. $$

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  • $\begingroup$ @Yang Thank you ! Can you recommend a good reference for such proof ? $\endgroup$
    – S. Maths
    Dec 7, 2018 at 10:32
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    $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Dec 7, 2018 at 16:45
  • $\begingroup$ 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 ? $\endgroup$
    – S. Maths
    Dec 7, 2018 at 17:22
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    $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Dec 7, 2018 at 17:39
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    $\begingroup$ It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. $\endgroup$
    – S. Maths
    Dec 7, 2018 at 19:31
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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.

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    $\begingroup$ 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. $\endgroup$
    – mme
    Dec 7, 2018 at 2:48
  • $\begingroup$ @Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book. $\endgroup$
    – S. Maths
    Dec 7, 2018 at 10:14
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    $\begingroup$ @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. $\endgroup$ Dec 7, 2018 at 14:35
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    $\begingroup$ @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. $\endgroup$
    – Deane Yang
    Dec 7, 2018 at 16:48
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Expanding @Piotr Hajlasz's answer, the result it true, and does appear in Warner's Foundations of differentiable manifolds and Lie groups. The result for the Laplacian appears as Theorem 6.32. Theorem 6.30 describes the local theory, then in 6.32 Warner describes how to apply the local theory to get a result for the whole manifold.

Warner does the local theory for periodic functions in $\mathbb{R}^n$ (with period $2\pi$). Then he uses charts which map to subsets of the $2\pi$ cube, and extends the result to the whole cube using freezing coefficients (essentially set the coefficients in your linear operator equal to constants outside some open set).

[Side notes: (1) Theorem 6.32 generalises easily to other elliptic second-order linear differential operators. (2) Warner doesn't discuss boundary points and boundary conditions, but Taylor does in Partial Differential Equations I, chapter 5.]

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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 $\rho=1, \delta=0$.

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