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My wife is reading "Functional Analysis and Applications" by Kesavan. On page $140$, at the bottom it says "Let $\Omega$ be an open set, $f\in L^2(\Omega)$, and $u\in H^1_0(\Omega)$ such that $-\Delta u+u=f$ in $\Omega$. And let $x_0 \notin \operatorname{sing.supp}(f)$."

Here $H_0^1(\Omega)$ is the closure of the test functions on $\Omega$ in $H^1(\Omega)=\{u\in L^2(\Omega)\mid\,D^\alpha u\in L^2(\Omega)\text{ for all } |\alpha|\leq 1\}$ and $\operatorname{sing.supp}(f)$ is the complement of the largest open set on which $f$ is $C^\infty$. The norm on $H^1(\Omega)$ is $$ \| u\|=\sum_{|\alpha|\leq 1}\|D^\alpha u\|_{_{L^2(\Omega)}}. $$

Why does such $x_0$ exist?

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If it doesn't then, by elliptic regularity, $u$ is smooth and a classical solution. – Jose27 Nov 22 '11 at 4:35
@Jose27 You could make your comment an answer. – Davide Giraudo Apr 22 '12 at 15:45

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