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 $90^\circ$ corner (as seen from inside), then again by direct computation, the integral is infinite. More precise statements can be made, depending on smoothness of $\partial E$.

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 $\partial E$.