Let $\Omega$ be a bounded domain, $f\in L^{\infty}(\Omega),$ and $0\leq u\in H_0^1(\Omega)$ is a non-negative solution of $\Delta u=f$. My question is as follows:

• Can we conclude that $u\in C^0(\bar{\Omega})$ without any boundary smoothness assumption on $\Omega$? If not, could you please provide a counterexample?

By employing De Giorgi iteration and the Harnack inequality, we can establish that $u$ belongs to $L^{\infty}(\Omega)$ and exhibits Hölder continuity in $\Omega$. Given that $u\in H_0^1(\Omega)$, my intuition is that $u$ remains continuous throughout the entire closure $\bar{\Omega}$ and equals zero on the boundary. However, since no assumptions have been made regarding the boundary $\partial \Omega$, I have been unable to prove this conjecture.