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We know the laplacean operator has a Green function which is smooth away from the boundary. Now, consider a linear operator of the form $\partial_i(a^{ij} \partial_j u)$.We can prove that this operator has a Green function as well. Now, if we just assume that $a^{ij} \in L^\infty$, and that $a^{ij}$ is uniformly elliptic, how much smoothness can be expected for the Green function?

Addendum: Well, we already know that away from the point on which the Dirac mass is supported, the solution is regular at least up to $C^{\alpha}$ by virtue of the theorem of De Giorgi-Nash-Moser.

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  • $\begingroup$ Look at the latest version of Gilbarg-Trudinger. As far as I know, it has the best results available. $\endgroup$
    – Deane Yang
    Commented Sep 28, 2012 at 2:32

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What you said is the most you can say about regularity away from the origin.

The only thing that can be added is the rate by which it goes to infinity at that point. There is an old result by Littman, Stampacchia and Weimberger which says that this rate is comparable with the corresponding one of the Green's function of the Laplacian.

Combining this with the local $C^\alpha$ estimates of De Giorgi, Nash and Moser you can get an estimate on how the $C^\alpha$ norm deteriorates when we approach the singularity.

I don't think anything else can be expected from the given assumptions.

This is the paper that I am referring to: Littman, W.; Stampacchia, G.; Weinberger, H. F. Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 43–77.

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