Balayage

This article is about mathematics. It is not to be confused with the hair painting technique.

Balayage is a French word meaning scanning or sweeping.

In potential theory, a mathematical discipline, balayage is a method devised by Henri Poincaré for reconstructing a harmonic function in a domain from its values on the boundary of the domain.^[1]

In modern terms, the balayage operator maps a measure μ on a closed domain D to a measure ν on the boundary ∂ D, so that the Newtonian potentials of μ and ν coincide outside D. The procedure is called balayage since the mass is "swept out" from D onto the boundary.

For x in D, the balayage of δ_x yields the harmonic measure ν_x corresponding to x. Then the value of a harmonic function f at x is equal to

f(x)=\int _{{\partial D}}f(y)\,d\nu _{x}(y).

References

↑ Solomentsev, E.D. (2001), "Balayage method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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