= Axial Poisson extension of a circular boundary patch
Take boundary data one on a disk of radius $a>0$ and zero outside. The <Poisson kernel for the upper half-space> gives on the axis
$$
\boxed{\phi(0,0,z)=z\int_0^a\frac{r\,dr}{(r^2+z^2)^{3/2}}=1-\frac{z}{\sqrt{a^2+z^2}},\qquad z>0.}
$$
The axis value tends to one at the boundary and behaves as $a^2/(2z^2)$ far away. The boundary data at the circular rim are understood almost everywhere: changing a value on the rim, a set of planar measure zero, leaves the integral unchanged. This does not assert a continuous boundary extension across the jump.
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