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.[1]

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),dnu _{x}(y).}

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.

• 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.