Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem $$ -\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega. $$ If $\varphi$ is a continuous function, then the problem is solvable for rather general $\Omega$. In fact, it suffices to assume that every point on the boundary $\partial \Omega$ is a endpoint of a segment, the other point of the segment lies outside $\Omega$. See the book 'Complex analysis' by Ahlfors.

Now we turn to the Possion equation $$ -\triangle u = \varphi, x \in \Omega, \quad u = 0, x \in \partial \Omega. $$ If $\Omega$ is a cube, we have the regularity result: $u$ is bounded in $W^{2,p}(\Omega)$ if $\varphi$ is bounded in $L^p(\Omega)$ with $ 1< p < \infty$.

This result can be extended to any $C^{1,1}$ domain $\Omega$. With some calculations, it seems that the result also holds for any convex $\Omega$. My questions are

(i)Does the regularity holds for more general $\Omega$, just like Dirichlet problem?

(ii)I have seen a paper which says that bad domain problem and bad coefficient problem has some relationship, could someone explain it more precisely?