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 &mu; on a closed domain D to a measure &nu; on the boundary &part; D, so that the Newtonian potentials of &mu; and &nu; coincide outside $$\bar D$$. The procedure is called balayage since the mass is "swept out" from D onto the boundary.

For x in D, the balayage of &delta;x yields the harmonic measure &nu;x corresponding to x. Then the value of a harmonic function f at x is equal to


 * $$ f(x) = \int_{\partial D} f(y) \, d\nu_x(y).$$