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 {\displaystyle \mu } on a closed domain D {\displaystyle D} to a measure ν {\displaystyle \nu } on the boundary ∂ D {\displaystyle \partial D} , so that the Newtonian potentials of μ {\displaystyle \mu } and ν {\displaystyle \nu } coincide outside D ¯ {\displaystyle {\bar {D}}} . The procedure is called balayage since the mass is "swept out" from D {\displaystyle D} onto the boundary. For x {\displaystyle x} in D {\displaystyle D} , the balayage of δ x {\displaystyle \delta _{x}} yields the harmonic measure ν x {\displaystyle \nu _{x}} corresponding to x {\displaystyle x} . Then the value of a harmonic function f {\displaystyle f} at x {\displaystyle 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).}
Examples
In gravity, Newton's shell theorem is an example. Consider a uniform mass distribution within a solid ball B {\displaystyle B} in R 3 {\displaystyle \mathbb {R} ^{3}} . The balayage of this mass distribution onto the surface of the ball (a sphere, ∂ B {\displaystyle \partial B} ) results in a uniform surface mass density. The gravitational potential outside the ball is identical for both the original solid ball and the swept-out surface mass. In electrostatics, the method of image charges is an example of "reverse" balayage. Consider a point charge q {\displaystyle q} located at a distance d {\displaystyle d} from an infinite, grounded conducting plane. The effect of the charges on the conducting plane can be "reverse balayaged" to a single "image charge" of − q {\displaystyle -q} at the mirror image position with respect to the plane.
References
Poincaré, H. (1890). "Sur les Equations aux Dérivées Partielles de la Physique Mathématique". American Journal of Mathematics. 12 (3): 211–294. doi:10.2307/2369620. ISSN 0002-9327. Poincaré, Henri (1899). Le Rot, Edouard; Vincent, Georges (eds.). Théorie du potentiel newtonien. Leçons professées à la Sorbonne pendant le premier semestre 1894–1895 [Theory of Newtonian potential. Lectures given at the Sorbonne during the first semester 1894–1895] (in French). Paris: Georges Carré et C. Naud. B. Gustafsson (2002). "Lectures on Balayage" (PDF). In Sirkka-Liisa Eriksson (ed.). Clifford Algebras and Potential Theory: Proceedings of the Summer School Held in Mekrijärvi, June 24–28, 2002. Report Series. Joensuu: University of Joensuu, Department of Mathematics. pp. 17–63. Retrieved 2025-03-02.