Boundary $W^{2,p}$ regularity needs more than Lipschitz boundary

The profile is harmonic and vanishes on the two sides. For $U(r,\theta )=r^{\gamma }\sin (\gamma \theta )$ one has, by [F2], $\mathrm{\Delta }U=[\gamma (\gamma -1)r^{\gamma -2}+\gamma r^{\gamma -2}-{\gamma }^2{r}^{\gamma -2}]\sin (\gamma \theta )=0,$ so $U$ is harmonic on the sector (in particular $U\in C^{\mathrm{\infty }}(\overline{{\mathrm{\Omega }}_{\omega }}\setminus \{0\})$). Moreover $U(r,0)=0$ and $U(r,\omega )=r^{\gamma }\sin (\pi )=0$, so $U$ vanishes on the two radial sides of ${\mathrm{\Omega }}_{\omega }$; at the reentrant vertex the sector has interior angle $\omega \in (\pi ,2\pi )$, so ${\mathrm{\Omega }}_{\omega }$ is a bounded Lipschitz domain that is not $C^{1,1}$ there.

The localized profile is in $H_0^1$. Since $\mathrm{\mid }U\mathrm{\mid }\le r^{\gamma }$ and $\mathrm{\mid }\mathrm{\nabla }U\mathrm{\mid }=\gamma r^{\gamma -1}$ (the gradient of the harmonic profile has absolute value $\gamma r^{\gamma -1}$ because the angular factor contributes a unit vector in polar coordinates), the integrals ${\int }_{r<1\mathrm{/}2}\mathrm{\mid }U{\mathrm{\mid }}^{2}dx$ and ${\int }_{r<1\mathrm{/}2}\mathrm{\mid }\mathrm{\nabla }U{\mathrm{\mid }}^{2}dx$ converge; hence $v=\zeta U\in H^1({\mathrm{\Omega }}_{\omega })$ with support in $\overline{B_{1\mathrm{/}2}}$. To approximate $v$ in $H^1$ by $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_{\omega })$ functions: first truncate radially, $v_{\epsilon }:=\chi (r\mathrm{/}\epsilon )v$ with $\chi \in C^{\mathrm{\infty }}$ equal to $0$ near $0$ and to $1$ for every $t\ge 2$. This transition need not be compactly supported: $v_{\epsilon }$ remains compactly supported because $v$ is supported in $\overline{B_{1\mathrm{/}2}}$. The estimate $\mathrm{\parallel }v-v_{\epsilon }{\mathrm{\parallel }}_{H^1}^2=O({\epsilon }^{2\gamma })$ follows from $\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2\lesssim r^{2\gamma -2}$ near the vertex. Next cut off in the angular variable with a smooth ${\eta }_{\delta }(\theta )$ vanishing for $\theta <\delta$ and for $\theta >\omega -\delta$ and equal to $1$ for $2\delta <\theta <\omega -2\delta$. For each fixed $\epsilon$, the squared $H^1$ error is $O(\delta )$, hence the $H^1$ norm error is $O({\delta }^{1\mathrm{/}2})$: near either side $U=O(r^{\gamma }\mathrm{dist}(\theta ,\{0,\omega \}))$, the angular cutoff derivative is $O({\delta }^{-1})$ on strips of angular width $O(\delta )$, and the resulting radial weight $r^{2\gamma -1}$ is integrable. The resulting functions are supported in a compact subset of the open sector and can be mollified there, so $v\in H_0^1({\mathrm{\Omega }}_{\omega })$ by definition of the closure.

The forcing is smooth and bounded, and the weak equation holds. Since $U$ is harmonic, $\mathrm{\Delta }v=\mathrm{\Delta }(\zeta U)=(\mathrm{\Delta }\zeta )U+2\mathrm{\nabla }\zeta \cdot \mathrm{\nabla }U$ on the sector; the right-hand side is supported in the annulus $\{r_0\le r\le 1\mathrm{/}2\}$ where $U$ and its gradient are smooth up to the two radial sides for $r\ge r_0>0$, so $f:=-\mathrm{\Delta }v\in C^{\mathrm{\infty }}({\mathrm{\Omega }}_{\omega })\cap L^{\mathrm{\infty }}({\mathrm{\Omega }}_{\omega })$ and hence $f\in L^p({\mathrm{\Omega }}_{\omega })$ for every finite $p$. For $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_{\omega })$ integration by parts on the compactly contained support gives $\int \mathrm{\nabla }v\cdot \mathrm{\nabla }\phi =\int (-\mathrm{\Delta }v)\phi =\int f\phi$; both sides are continuous in $\phi$ in the $H^1$ norm, so the identity holds for every $\phi \in H_0^1({\mathrm{\Omega }}_{\omega })$ by [F1]: $v$ is a weak solution of $-\mathrm{\Delta }v=f$ with zero boundary values.

The second derivatives diverge exactly above the threshold. On $\{r<r_0\}$ one has $v=U$, a function homogeneous of degree $\gamma$; write $U=\mathrm{Im}{z}^{\gamma }$ on the sector branch. Direct differentiation gives $U_{11}=\gamma (\gamma -1)r^{\gamma -2}\sin ((\gamma -2)\theta )$, $U_{12}=\gamma (\gamma -1)r^{\gamma -2}\cos ((\gamma -2)\theta )$ and $U_{22}=-U_{11}$. Thus the Frobenius Hessian norm is $\sqrt{2}\gamma (1-\gamma )r^{\gamma -2}$, so there are constants $0<c_1\le c_2<\mathrm{\infty }$ with $c_1{r}^{\gamma -2}\le \mathrm{\mid }{D}^{2}v\mathrm{\mid }\le c_2{r}^{\gamma -2}$ on ${\mathrm{\Omega }}_{\omega }\cap \{0<r<r_0\}$ (by the displayed nonvanishing norm, since $0<\gamma <1$). By [F2], ${\int }_{{\mathrm{\Omega }}_{\omega }\cap B_{r_0}}\mathrm{\mid }{D}^{2}v{\mathrm{\mid }}^{p}dx\asymp {\int }_0^{r_0}{r}^{p(\gamma -2)+1}dr,$ which by [F4] diverges exactly when $p(\gamma -2)+1\le -1$, that is $p\ge \frac{2}{2-\gamma }$. Using $\gamma =\pi \mathrm{/}\omega$ gives $\frac{2}{2-\gamma }=\frac{2\omega }{2\omega -\pi }$, so the weak solution $v$ is not in $W^{2,p}({\mathrm{\Omega }}_{\omega })$ for those exponents; taking $p=2$ (which is allowed because $\gamma <1$) shows in particular that $v\mathrm{\notin }{H}^2({\mathrm{\Omega }}_{\omega })$.

Conclusion. On the bounded Lipschitz reentrant domain ${\mathrm{\Omega }}_{\omega }$ there is a weak solution $v\in H_0^1({\mathrm{\Omega }}_{\omega })$ of $-\mathrm{\Delta }v=f$ with $f\in C^{\mathrm{\infty }}\cap L^{\mathrm{\infty }}$, which fails to lie in $W^{2,p}({\mathrm{\Omega }}_{\omega })$ for every $p\ge 2\omega \mathrm{/}(2\omega -\pi )$. Thus smooth data and a Lipschitz boundary alone do not guarantee weak-solution $W^{2,p}$ regularity. This example is not a counterexample to the a priori estimate Global $W^{2,p}$ Dirichlet estimate on a $C^{1,1}$ domain, whose domain already assumes $u\in W^{2,p}$; it makes no claim that the estimate’s $C^{1,1}$ boundary hypothesis is necessary.