Balayage
In potential theory, a mathematical discipline, balayage (from French: balayage "scanning, sweeping") is a method devised by Henri Poincaré for reconstructing an harmonic function in a domain from its values on the boundary of the domain.
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
¯
{\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 δx yields the harmonic measure νx corresponding to x. Then the value of a harmonic function f at x is equal to
f
(
x
)
=
∫
∂
D
f
(
y
)
d
ν
x
(
y
)
.
{\displaystyle f(x)=\int _{\partial D}f(y)\,d\nu _{x}(y).}
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