Balayage

In potential theory, a mathematical discipline, balayage (from French: balayage "scanning, sweeping") 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]

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

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 ${\displaystyle {\bar {D}}}$. 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

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

References

1. Solomentsev, E.D. (2001) [1994], "Balayage method", Encyclopedia of Mathematics, EMS Press