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G. Blaickner
G. Blaickner
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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 C^{\infty}_{c}(M)$$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)

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 $f\in C^{\infty}_{c}(M)$. 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)

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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G. Blaickner
G. Blaickner
  • 1.4k
  • 4
  • 16

On elliptic operators on non-compact manifolds

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 $f\in C^{\infty}_{c}(M)$. 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)