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Weyl's lemma: A function $u \in L_{loc}^1(\Omega)$ is harmonic if $$\int_\Omega u \Delta \varphi = 0 \quad \forall \varphi \in C_c^\infty(\Omega) $$ Here $\Omega$ is a domain i.e open and connected.

What I find confusing is the first step of the proof: Consider a radial family of mollifiers $\rho_\epsilon$, and $u_\epsilon = u \ast \rho_\epsilon$ $$\int_\Omega u_\epsilon \Delta \varphi = \int_\Omega u(\Delta \varphi \ast \rho_\epsilon) = \int_\Omega u \Delta (\varphi \ast \rho_\epsilon) = 0 \quad \forall \varphi \in C_c^\infty(\Omega_\epsilon) $$

Where $\Omega_\epsilon = \{ x\in \omega: dist(x, \partial \Omega) > \epsilon \}$. In particual $\Delta u_\epsilon = 0$ in $\Omega_\epsilon$

That last part is what I don't understand, how does $$\int_\Omega u_\epsilon \Delta \varphi = 0 \quad \forall \varphi \in C_c^\infty(\Omega_\epsilon) $$ implies that $\Delta u_\epsilon = 0$ in $\Omega_\epsilon$ ?

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  • $\begingroup$ Did you try to integrate by parts? $\endgroup$
    – fwd
    Feb 11, 2023 at 21:28
  • $\begingroup$ I don't know the lemma, but it seems like you want the integral $=0?$ Or do you want the integral to just exist? $\endgroup$ Feb 11, 2023 at 21:46
  • $\begingroup$ ... indeed : look at the way you have written the first "equation". $\endgroup$
    – Jean Marie
    Feb 11, 2023 at 21:49
  • $\begingroup$ I don't think one can integrate by parts, at least in the book, the set $\Omega$ is just assumed to be open and connected, there's no regularity assumptions on $\partial \Omega$ $\endgroup$
    – Franlezana
    Feb 12, 2023 at 0:42
  • $\begingroup$ I think it's possible to integrate by parts over $\Omega_\epsilon$ $\endgroup$ Feb 12, 2023 at 0:46

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As suggested in the comments, you want to integrate by parts. From what I can tell, your worry is that $\Omega$ is only assumed to be open and bounded, and integration by parts requires some kind of regularity assumption on the boundary. You're right, but this is not an issue here because $\varphi $ has compact support in $\Omega_\varepsilon$. Indeed, for each $\varphi \in C^\infty_0(\Omega_\varepsilon)$, choose a set $U \subset \Omega$ such that $\operatorname{supp} \varphi \subset \subset U \subset \subset \Omega_\varepsilon$ and $\partial U$ is smooth. Then, by Green's identity\begin{align*} \int_{\Omega_\varepsilon} \varphi \Delta u_\varepsilon \, dx &=\int_U \varphi \Delta u_\varepsilon \, dx \\ &= \int_U u_\varepsilon \Delta \varphi \, dx + \int_{\partial U}\bigg(u_\varepsilon \frac{\partial \varphi }{\partial \nu} - \varphi \frac{\partial u_\varepsilon}{\partial \nu} \bigg)\, d\mathcal H^{n-1}_x\\ &=0. \end{align*} Note that all the derivative on $u_\varepsilon$ make sense because $u_\varepsilon$ is smooth. Since $\varphi$ was arbitrary, it follows that $\Delta u_\varepsilon =0$ in $\Omega_\varepsilon$ as required.

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