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Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (compactly-supported) smooth sections. Now, lets say I have given an elliptic linear differential operator $D:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$. I am interested in the problem of finding solutions $u\in L^{2}(M)$ to the problem $$Du=f$$ for a compactly-supported source $f\in \Gamma^{\infty}_{c}(M)$. Of course, if $M$ is compact everything is clear (cf. Fredholm theory), so I am asking specifically for the non-compact case (Lets say $M$ is complete, to be concrete).

Of course, the question is very much related to the question whether a Green's function exists for $D$. I am aware of the paper of Malgrange (see 1. below), which states that any elliptic operator with constant-coefficient on $\mathbb{R}^{n}$ admits a Green's function, however, the original paper is in French and hence not accessible to me. A textbook version of this statement should also be somewhere contained in Hörmander, but only in the Euclidean case. There is this paper by Li-Tam (see 2. below), which give a more constructive proof (from a more differential geometric point of view) of Malgrange's theorem in the specific case of the Laplace-Beltrami operator. They do, however, state in the introduction of their paper that

In 1955 Malgrange studied elliptic operators on complete Riemannian manifolds which satisfied unique continuation property. In particular, he proved that the Laplace operator admits a symmetric Green's function [...].

But again, the original paper of Malgrange is not accessible to me and it is, as far as I know, it is for $\mathbb{R}^{n}$. So, let me summarise:

Does anyone know a reference regarding the existence of $L^{2}$-solutions of elliptic problems on non-compact manifold with sources in $C^{\infty}_{c}$? In particular, what are the precise assumptions on the manifold (completeness I guess, any curvature bounds?) and operator (unique continuation property?)

  1. Malgrange: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier (Grenoble), 6:271-355, 1955/6. (see here for an online version at Numdam)
  2. Li, Tam: Symmetric Green's Functions on Complete Manifolds, American Jounral of Mathematics, 109(6):1129-1154, 1987. (See here for an online version at jstor)
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    $\begingroup$ FYI, the general results in Malgrange's article don't depend on any Riemannian metric, so there is no natural $L^2(M)$ space on them. The closest result that might be of interest to you seems to be Thm.III.5: $D\colon H_{loc}^k(M) \to H_{loc}^{k-m}(M)$ is surjective, where the function spaces are local Sobolev ($k=\pm \infty$ are allowed), $m$ is the degree of $D$, and $D$ is both elliptic and has analytic coefficients. He later shows that the existence of a fundamental solution follows, but again without any global $L^2(M)$ properties. $\endgroup$
    – Igor Khavkine
    Commented Jul 30 at 23:56
  • $\begingroup$ @IgorKhavkine Thanks a lot for your comment! Thats indeed interesting, but unfortunately, not quite what I am looking for. I am a bit suprised that there seems to be no general result. At least for the Poisson equation on complete manifold, I think I have seen the claim somewhere that there is a Green's function $G:C^{\infty}_{c}(M)\to L^{2}(M)$, but I might be wrong though. $\endgroup$
    – G. Blaickner
    Commented Aug 1 at 7:05
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    $\begingroup$ I don't have a copy of the book at hand to check myself, but you may try Eichhorn's "Global analysis on open manifolds". $\endgroup$
    – Tobias Diez
    Commented Aug 1 at 12:12

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Too long for a comment. I would say that your problem is a semi-global solvability question. You will find in chapter 26 of Hörmander’s ALPDO a precise definition for that property which suits well the fact that $f$ is compactly supported. In that chapter, the author deals with scalar differential operators and proves a quite general solvability result under much weaker assumptions than ellipticity.

In your case, ellipticity will provide immediately some a priori estimates where the $L^2$ norm of $D^*\phi$ dominates the $ H^m$ norm of $\phi$ with a constant depending only on the support of $\phi$ ($m=$the order of $ D $ and $\phi$ is smooth compactly supported.)

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