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I have again something from Stein-Shakarchi I would really appreciate some help with. Any references are also welcome!

Suppose $L$ is a linear partial differential operator with constant coefficients. Show that when $d \geq 2$, the linear space of solutions $u$ of $L(u)=0$ with $ u \in C^{\infty}(\mathbb{R}^{d})$ is not finitely dimensional.

Thanks in advance!

EDIT: $L$ takes the form $$L= \sum_{|\alpha| \leq n}{a_{\alpha}\left(\frac{\partial{}}{\partial{x}}\right)^{\alpha}}$$ with $a_{\alpha} \in \mathbb{C}$ constants.

  • 0
    How is a partial differential operator defined?2012-05-02
  • 0
    Edited. Sorry for not being thorough.2012-05-02

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