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I have $u \in H^1_0(\Omega)$ where $\Omega$ is bounded which solves some elliptic PDE of the form: $$-\Delta u + h(u) = f$$ in $\Omega$, where $f \in L^2(\Omega)$

$$u = 0$$ on $\partial\Omega$, say.

How do I get an estimate for $\lVert \Delta u \rVert_{L^2(\Omega)}$? Surely I can't even write that because $u$ is only in $H^1$? But apparently I can. Can I just do something with the strong form?

Thanks for any help.

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Isn't the Sobolev norm $$\left\Vert \phi\right\Vert _{H^{1}}^{2}=\left\Vert \phi\right\Vert _{L^{2}}^{2}+\left\Vert \nabla\phi\right\Vert _{L^{2}}^{2} ?$$ so surely if $\Delta u\in H^{1}$ then $\Delta u\in L^{2}$. –  Anthony Carapetis Apr 18 '12 at 11:00
    
@Pox but how is $\Delta u \in H^1$? –  quiop Apr 18 '12 at 11:10
    
...ignore me, I thought you wrote $\Delta u$ is in $H^1$. –  Anthony Carapetis Apr 18 '12 at 11:21
    
@DavideGiraudo suppose that $h \equiv 0.$ –  quiop Apr 18 '12 at 19:19
    
Where does the problem come from? –  Davide Giraudo Apr 19 '12 at 21:22

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