The Poisson kernel is a fundamental concept in harmonic analysis and potential theory, particularly in the study of solutions to the Laplace equation. It is used, among other things, to express the solution to the Dirichlet problem for the unit disk.
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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 , 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.