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Given $ f \in L^2(U)$ and $\partial U$ is $C^2$. If $u \in H^1_0(U)$ is the unique weak solution to a simple elliptic problem with Dirichlet boundary condition $$ \begin{cases} - \Delta u + \mu u = f, \text{ in } \Omega \\ u = 0, \text{ in } \partial \Omega. \end{cases} $$ We have that $u \in H^2(U)$ and we have the estimate $$ \|u\|_{H^2(U)} \leq C\|f\|_{L^2(U)}. $$

I know that a more general version of this proof is presented in Evans, but I would like to write it only for the specific case that I need. I would assume that this would simplify the proof but I can't seem to do it.

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  • $\begingroup$ You could try to do the same estimate for newtonian kind of potential and then subtract off the newtonian potential from the given function to get usual harmonic equation like situation? $\endgroup$ Commented Jan 29 at 6:18
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    $\begingroup$ Write that $-\Delta u=g$ with $g=f-\mu u\in L^2$, so that essentially it suffices to go over the proof for the Lapace equation. And simply adapt Evans' proof whenever possible, given that you are now dealing with the basic Laplacian instead of more general elliptic operators? $\endgroup$ Commented Jan 29 at 10:38

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