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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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    If $h \equiv 0$, then the problem is trivial - just $H^2$ estimates for the Poisson problem. In general, it depends on $h$.2013-04-14

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