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Let $M := C_0^{\infty}(\mathbb{R}^n)$ denote the smooth maps with compact support. Then we have a map

$\Delta:M\rightarrow M,\,\, f\mapsto \Delta f$,

where $\Delta f = \sum_{i=1}^{n} \frac{\partial^2}{\partial x_i^2}f$ is the Laplacian. I am wondering if $\Delta$ is surjective, i.e. if for any $f\in M$ there exists an $F\in M$ with $\Delta F = f$. Is that true?

Thanks for your help!

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    re LVK's comment , start with $\mathbb R^1$2012-09-13

2 Answers 2

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It is very far from being surjective. Note that if $f\in C^\infty_0$ and $u$ is any harmonic function in the entire space, then $\int (\Delta f)u=\int f(\Delta u)=0$ (integration by parts or Green). This imposes infinitely many independent restrictions on the functions that can be represented as Laplacians of smooth compactly supported functions in every dimension above $1$ (in dimension $1$ the only harmonic functions are linear).

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    Ah - yeah. Got it now. Thank you!2012-09-13
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This can not be true. If it would, it would imply that the fundamental solution of laplace equation is unique, which is obviously false. Igor.

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    See also: http://www.amazon.de/Numerical-Schemes-Conservation-Advances-Mathematics/dp/35190272082012-09-13