We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$.

Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, consider the problem in a domain $\Omega$ (in my question a planar domain) \begin{equation} (1)\qquad \left\lbrace \begin{array}\mbox{div}\left[(\epsilon+|\nabla v_\epsilon|^2)^{(p-2)/2}\nabla v_\epsilon\right]=0\mbox{ in }\Omega\\ v_\epsilon=g\mbox{ on }\partial\Omega\end{array}\right. \end{equation} Then, if the domain is regular "enough", we get that the solution $v_\epsilon$ is $C^{2,\rho}$ and it converges strongly to the solution of \begin{equation} (2)\qquad\left\lbrace \begin{array} \mbox{div}(|\nabla v|^{p-2}\nabla v) =0\mbox{ in }\Omega\\ v=g\mbox{ on }\partial\Omega\end{array}\right. \end{equation}

First, we can find this result in Gilbarg/Trudinger book, Chapter 15. Theorems 15.11, 15.18, 15.19.

In the book, this result works if the domain satisfies the exterior sphere condition. Therefore, the domain should be at least $C^2$ (I guess Ziemer's book about regularity says that it is enough $C^{1,\alpha}$).

(A) if we have a rectangular domain (in the plane), is this result still true? I mean, the regularity of solutions of (1) is still $C^{2,\rho}$?

(B) if (A) holds, then can we think about the problem in a $C^1$ by parts domain? For instance, one can think about the domain $$ \Omega=\lbrace 0<x<1, 0<y<h(x)\rbrace $$ where $h$ is a $1$-periodic, $C^1$ function.