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Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:

  • The fundamental solution $\Gamma(x)$ of $L$;
  • And the green kernel $\mathcal G(x)$of $L$.

In which case, we have $\Gamma(x)=\mathcal G(x)$ or the contrary ?

Thank you in advance.

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    $\begingroup$ isn't it just that the fundamental solution does not respect the boundary condition, while the Green function does? $\endgroup$
    – Carlo Beenakker
    Commented Sep 7, 2021 at 8:47
  • $\begingroup$ Can we say so, without boundary condition, we have: Fundamental solution=Green kernel? $\endgroup$
    – Z. Alfata
    Commented Sep 7, 2021 at 9:38
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    $\begingroup$ yes, that is my understanding. $\endgroup$
    – Carlo Beenakker
    Commented Sep 7, 2021 at 10:19
  • $\begingroup$ The Wikipedia article on fundamental solution says: In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function, which normally further addresses boundary conditions. Thank you @Beenakker $\endgroup$
    – Z. Alfata
    Commented Sep 7, 2021 at 11:57

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