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I know that there are a lot of (great) books treating regularity of weak solutions of elliptic pdes (such as Gilbarg-Trudinger), but what about regularity of very weak solutions, that is, solutions in the distributional sense? For concreteness, consider a bounded domain $\Omega \subset \mathbb{R}^2$ with smooth boundary, and two continuous functions $u, f: \Omega \to \mathbb{R}$ satisfying $-\Delta u = f$ on $\Omega$ in the distributional sense, that is $$ -\int_{\Omega}{u\Delta \phi \mathrm{d}x} = \int_{\Omega}{f \phi\mathrm{d}x}\quad \forall \phi \in \mathscr{C}^{\infty}_{\text{c}}(\Omega). $$ I've read in many articles statements like "if $u \in L^{\infty}_{\text{loc}}(\Omega)$, then by standard elliptic regularity $u \in \mathscr{C}^{1, \alpha}_{\text{loc}}(\Omega)$". What do they mean with standard elliptic regularity in this case? Any help would be appreciated, even just a reference.

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    Just a remark: weak solutions are solutions the have weak derivatives. In other words, the distributional derivatives must be (square) integrable.2012-08-17
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    Uhm, I think this should read "if $f\in L^\infty_{loc}$, then". In case this is true this is basically covered by Gilbarg Trudinger, Theorem 8.34.2012-08-20
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    Was it a bounty question? First time in my life I got awarded 500 points!2012-08-23

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