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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 $y\in \partial E$?

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    $\begingroup$ The answer depends on how smooth your domain is. In dimension 2, if very smooth, it is finite, but if it can have, say 90 degree corner, then the integral can be infinite. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 29, 2013 at 2:37

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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 $\partial E$.

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