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.^[1]

In modern terms, the balayage operator maps a measure ${\displaystyle \mu }$ on a closed domain ${\displaystyle D}$ to a measure ${\displaystyle \nu }$ on the boundary ${\displaystyle \partial D}$, so that the Newtonian potentials of ${\displaystyle \mu }$ and ${\displaystyle \nu }$ coincide outside ${\displaystyle {\bar {D}}}$. The procedure is called balayage since the mass is "swept out" from ${\displaystyle D}$ onto the boundary.

For ${\displaystyle x}$ in ${\displaystyle D}$, the balayage of ${\displaystyle \delta _{x}}$ yields the harmonic measure ${\displaystyle \nu _{x}}$ corresponding to ${\displaystyle x}$. Then the value of a harmonic function ${\displaystyle f}$ at ${\displaystyle x}$ is equal to$${\displaystyle f(x)=\int _{\partial D}f(y)\,d\nu _{x}(y).}$$

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 ${\displaystyle B}$ in ${\displaystyle \mathbb {R} ^{3}}$. The balayage of this mass distribution onto the surface of the ball (a sphere, ${\displaystyle \partial B}$) 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 ${\displaystyle q}$ located at a distance ${\displaystyle d}$ 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 ${\displaystyle -q}$ at the mirror image position with respect to the plane.