= Poisson kernel
{c}
{title2=$P_\Omega(x,\xi)=-\partial_{n_\xi}G_\Omega(x,\xi)$}
{wiki}
A Poisson kernel represents the <harmonic extension> of <Dirichlet boundary data> by an integral over the boundary. When the <Dirichlet Green function> is normalized by $-\Delta G_\Omega=\delta$, its negative outward normal <derivative> gives the displayed kernel. The <Poisson kernel for the upper half-space> and the <Poisson kernel for the upper half-plane> give explicit examples with different boundary dimensions. Boundedness or other stated growth conditions are needed when selecting a solution in an unbounded domain.
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