Balayage

This article is about the method in mathematics. For the hair painting technique, see Hair highlighting § Hair painting.

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

Examples

The field of a positive charge above a flat conducting surface, found by the method of images.

In gravity, Newton's shell theorem is an example. Consider a uniform mass distribution within a solid ball $B$ in $\mathbb {R} ^{3}$. The balayage of this mass distribution onto the surface of the ball (a sphere, $\partial B$) 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 $q$ located at a distance $d$ 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 $-q$ 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.