The reentrant sector singularity has an explicit Sobolev threshold

Given: $\omega \in (\pi ,2\pi )$, the sector $S_{\omega }$ above, $\alpha =\pi \mathrm{/}\omega \in (0,1)$, and $u(r,\theta )=r^{\alpha }\sin (\alpha \theta )$.

A class in $H^1(S_{\omega })$ is a local weak solution of $-\mathrm{\Delta }u=0$ on $S_{\omega }$ if ${\int }_{S_{\omega }}\mathrm{\nabla }u\cdot \overline{\mathrm{\nabla }v}dx=0$ for every $v\in C_c^{\mathrm{\infty }}(S_{\omega })$. (Local weak solutions of a divergence-form operator)

For every $C^2$ function $w$ on the punctured plane, the chain rule gives $\mathrm{\Delta }w=w_{rr}+r^{-1}{w}_r+r^{-2}{w}_{\theta \theta }$ in polar coordinates; in particular $\mathrm{\Delta }(r^{\alpha }\sin (\alpha \theta ))=0$ on the sector. (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, Weak derivative of a locally integrable function)

Polar integration on the sector is ${\int }_{S_{\omega }}fdx={\int }_0^{\omega }{\int }_0^{1}f(r\cos \theta ,r\sin \theta )rdrd\theta$. (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma)

A class $u$ lies in $H^m(S_{\omega })$ for an integer $m\ge 0$ exactly when all its weak partial derivatives of order $\le m$ lie in $L^2(S_{\omega })$; the weak derivatives of the smooth function $u$ on $S_{\omega }\setminus \{0\}$ are the classical ones, and on the sector $r>0$ the classical derivatives $D^{j}u$ are bounded by constant multiples of $r^{\alpha -j}$. Along each fixed ray $x=r{e}_r$, the radial derivative satisfies ${\mathrm{\partial }}_r^{m}u(r,\theta )=D^{m}u(r{e}_r)[e_r,\dots ,e_r]$; since $\mathrm{\mid }{e}_r\mathrm{\mid }=1$, if all Cartesian derivatives of order $m$ lie in $L^2$, then this radial derivative also lies in $L^2$. (Integer-order Sobolev spaces and their norms, Weak derivative of a locally integrable function)

The reentrant sector is a bounded Lipschitz domain that fails the $C^1$ boundary-chart condition at its vertex, and $u$ is a local weak solution of $-\mathrm{\Delta }u=0$ on it with $u\in H^1(S_{\omega })\setminus H^2(S_{\omega })$. (Boundary $H^2$ regularity needs domain regularity)

Let $w(r,\theta )=r^{\gamma }\mathrm{\Psi }(\theta )$ on this sector, where $\gamma >-1$ and $\mathrm{\Psi }$ is smooth on $[0,\omega ]$. For $0<t<1$, the intrinsic seminorm ${\int }_{S_{\omega }}{\int }_{S_{\omega }}\frac{\mathrm{\mid }w(x)-w(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{2+2t}}dxdy$ is finite if $t<\gamma +1$. To see this, put $A_j=\{2^{-j-1}<r<2^{-j},0<\theta <\omega \}$ and ${\lambda }_j=2^{-j}$. The angular formula extends smoothly to a slightly larger interval because $\omega <2\pi$, so near pairs in comparable shells satisfy $\mathrm{\mid }w(x)-w(y)\mathrm{\mid }\le C{\lambda }_j^{\gamma -1}\mathrm{\mid }x-y\mathrm{\mid }$; integrating such pairs gives $C{\lambda }_j^{2\gamma +2-2t}$. Separated pairs in comparable shells give the same bound from $\mathrm{\mid }w\mathrm{\mid }\le C{\lambda }_j^{\gamma }$ and $\mathrm{\mid }x-y\mathrm{\mid }\ge c{\lambda }_j$. For noncomparable shells $A_j,A_k$ with $k\ge j+2$, $\mathrm{\mid }x-y\mathrm{\mid }\ge c{\lambda }_j$; integrating the two terms ${\lambda }_j^{2\gamma }$ and ${\lambda }_k^{2\gamma }$ in $\mathrm{\mid }w(x)-w(y){\mathrm{\mid }}^2$ and summing over $k$ gives at most $C{\lambda }_j^{2\gamma +2-2t}$, with the inner sum geometric since $\gamma >-1$. The final sum over $j$ converges exactly when $t<\gamma +1$.

For $Du=\mathrm{\nabla }u=\alpha r^{\alpha -1}(\sin ((\alpha -1)\theta ),\cos ((\alpha -1)\theta ))$, choose two small disjoint balls $A,B$ compactly contained in $\{1\mathrm{/}2<r<1,0<\theta <\omega \}$, centered at the same radius and at angles $\omega \mathrm{/}4$ and $3\omega \mathrm{/}4$. Their gradient values differ because the direction-angle difference is $(\alpha -1)\omega \mathrm{/}2=(\pi -\omega )\mathrm{/}2\mathrm{\ne }0$; shrinking the balls gives $\mathrm{\mid }Du(x)-Du(y)\mathrm{\mid }\ge c>0$ on $A\times B$. By homogeneity $Du(\lambda x)={\lambda }^{\alpha -1}Du(x)$, the order-$t$ seminorm integral over $2^{-j}A\times 2^{-j}B$ is at least $c^{\mathrm{\prime }}{2}^{-j(2\alpha -2t)}$. These product sets are pairwise disjoint as $j$ varies, so their sum diverges for $t\ge \alpha$.