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Out of curiosity, I was wondering why in the analysis of elliptic PDEs with "source" in $H^{-1}(\Omega)$ everyone writes the right-hand side in the form $$g_0- \operatorname{div} g$$ with $g_0,g\in L^2(\Omega)$ instead of the (equivalent but apparently) more precise $$f-\operatorname{div} (\nabla f)$$ with $f$ in $H_0^1(\Omega)$.

Any real advantage behind the first choice?

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    $\begingroup$ By Riesz representation theorem, both expressions span the whole of $H^{-1}$. There is no difference between the two in terms of generality. They give two equivalent representations of $H^{-1}$. The second one is exactly the one that one gets by Riesz representation theorem. $\endgroup$
    – Kosh
    Commented Aug 14, 2020 at 1:13
  • $\begingroup$ If you use the second representation, it gets awkward to prove that a distribution is in $H^{-1}$. The first one tells you that any derivative of an $L^2$ function is in $H^{-1}$, no special structure required $\endgroup$
    – Piero D'Ancona
    Commented Aug 14, 2020 at 8:37
  • $\begingroup$ Note sure to get your point. Let me try to explain better what I meant with the post. If one is interested in the properties of equations of the type $L u = f$ with $L$ the classical 2nd order linear elliptic operator then, from my point of view, the first RHS tells that the source is the sum of an $L^2$ function and the divergence of an $L^2$ function, while the second tells that it is the sum of a "more regular" $H_0^1$ function with the divergence of an $L^2$ function which one knows being "regular" because is the gradient of an $H_0^1$ function. $\endgroup$
    – Kosh
    Commented Aug 14, 2020 at 9:01
  • $\begingroup$ So, it seems like the 2nd one reveals more structure. Also, it is the gradient of the same function $f$ so that even from the notational point of view, one does not need to introduce two functions. If one decides to write a book, I cannot find good reasons to use the first one instead of the second one, other than the willingness of following the tradition. $\endgroup$
    – Kosh
    Commented Aug 14, 2020 at 9:02
  • $\begingroup$ Maybe, an advantage is that if one derives estimates in terms of $g_0$ and $g$, then one can immediately understand what happens for "source" in the more regular space $L^2$ by simply forgetting about $g$. $\endgroup$
    – Kosh
    Commented Aug 14, 2020 at 13:10

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