Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic function taking boundary value $F(x,\cdot)$. Poisson's kernel is defined as $P(x,y)=-\frac{\partial}{\partial n(y)}G(x,y)$ with $x\in E$ and $y\in \partial E$ and $n(y)$ is the outer normal at $y$.
Question: is $\int_E P(x,y)\,dx$ always finite for all $y\in \partial E$?
In dimension $2$, let $U$ be the unit disc, and $f:U\to E$ a conformal map. Then the integral in question is $$\int_U P_U(z,1)|f'(z)|^2dA,$$ where $dA$ is the area element in $z$-plane. When $f$ is the identity map, the integral converges by direct computation. If $E$ is sufficiently smooth, $f'$ is bounded, and the integral is finite. But if for example, $E$ has a corner $\leq90^\circ$ (as seen from inside), then again by direct computation, the integral is infinite. More precise statements can be made, depending on smoothness of $\partial E$.
