Bdtyktl

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 $uin H^1(M)$, and $fin L^2(M)$ 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.

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.

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
$$
Delta u = (delta^{ij} + a^{ij}(x))partial^2 + b^kpartial_ku
$$
where $|a^{ij}|, |b_k| < epsilon << 1$.
Therefore, if $Delta_g u = f$, then
$$
Delta_0u = -a_{ij}partial^2_{ij}u - b^kpartial_ku + f
$$
Therefore, by $(*)$
$$
|u|_{H^2} le C(epsilon |u|_{H^2} + |f|_{L^2}).
$$
If the neighborhood is sufficiently small, then $Cepsilon < 1$ and therefore,
$$
|u|_{H^2} le C|f|_{L^2}.
$$