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]
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 δx yields the harmonic measure νx 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) [1994], "Balayage method", Encyclopedia of Mathematics, EMS Presss
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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