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]

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 ${\bar {D}}$ . 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

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

References

Solomentsev, E.D. (2001) [1994], "Balayage method", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Solomentsev, E.D. (2001) [1994], "Balayage method", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4