I am looking for an explicit example that can be verified that demonstrates this: there is a Lipschitz domain $\Omega$ such that given $f \in L^2(\Omega)$, the PDE $$-\Delta u = f \text{ on $\Omega$}$$ $$u = 0 \text { on $\partial\Omega$}$$ has a solution $u \in H^1_0(\Omega)$ with $\Delta u \in L^2(\Omega)$ but $u \not\in H^2(\Omega)$.
Preferably the example doesn't use polar coordinates. Does anyone have a reference? Thanks