It is well known that there exist counterexamples to the standard elliptic regularity theorem (sometimes called the "shift theorem") for second order differential equations in the case of Lipschitz domains. For example, one can construct a Lipschitz domain $\Omega$ and function $u$ such that $\Delta u(x) = f(x)$ with $f \in L^2(\Omega)$ and the trace of $u$ continuous but $u \notin H^2(\Omega)$.
I am reading Jost's book on elliptic PDEs (and Gilbarg and Trudinger in parallel, although I find Jost's book easier). It seems to me that the results in those books imply that
$$w(x) = \int_\Omega \log|x-y| f(y) dy$$
is in $H^2(\Omega)$ when $f \in L^2(\Omega)$ for generic open bounded sets $\Omega$, and that a bound of the form
$$\|w\|_{H^2(\Omega)} \leq C \|f\|_{L^2(\Omega)}$$
holds. Unless I'm mistaken, this is a consequence of Theorem 12.1.1 in Jost. This is surprising to me. I would expect to require assumptions on the regularity of the boundary to get this. Have I misread something? Does one require some regularity of $\partial\Omega$ for this result?
Similarly, my reading of Jost Theorem 13.1.1 is that $w \in C^{2,\alpha}\left(\overline{\Omega}\right)$ when $f \in C^{0,\alpha}\left(\overline{\Omega}\right)$ and $0 <\alpha < 1$. Again, this is for a generic open bounded set $\Omega$. This seems really surprising! Doesn't one need boundary regularity to get something like this?
Am I misreading Jost or does one generally get stronger regularity for the Newton potential than one does for the solutions of the Poisson equation on the same domain?