Question stated roughly: Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one approach solving a quasi-linear or non-linear elliptic Dirichlet problem with rough data?

Evidence: Existence of Poisson kernels guarantee existence of solutions to the Laplace equation even with rough boundary. In the interior nevertheless, we have infinite smoothness. It's true that laplacian has all the nice properties, nevertheless, one might hope to be able to obtain similar, but possibly weaker, results in more general classes of equations.

The case of linear equations: Let us consider the case of a ball, $B$. In case we have a linear equation in the `divergence' form, the usual Hilbert space approach theory we get existence of solutions so long as the boundary data lies in the image of the trace operator, $\operatorname{Tr}: H^1(B)\rightarrow L^2(\partial B)$. Equivalently, if the boundary data can be extended to an $H^1$ function in the interior, the standard theory yields a generalised solution in $H^1$.

Nevertheless, if the equation is not in divergence form, this approach will not work. Therefore, one might restrict oneself to the case where the standard Schauder theory works: divergence equations with co-efficients in a H\"older space. The problem with the conventional Schauder theory is that it only works when the boundary data is $C^{2,\alpha}$. What I have been able to find has been an article by Gilbarg and H\"ormander which extends the regularity and existence results to less regular boundary data (actually with less restriction even on the co-efficients).

Question: Are there major approaches other than weighted H\"older spaces to this problem? In case the equation is fully non-linear, say $F(x,D^2 u)=0$, is it reasonable to expect, under certain structural conditions on the operator, to be able to prove existence and regularity in appropriate weighted H\"older spaces? Are there any major results in these directions?