Double layer potential

In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions. Thus a double layer potential u(x) is a scalar-valued function of x ∈ R³ given by

$u({\mathbf {x}})={\frac {-1}{4\pi }}\int _{S}\rho ({\mathbf {y}}){\frac {\partial }{\partial \nu }}{\frac {1}{|{\mathbf {x}}-{\mathbf {y}}|}}\,d\sigma ({\mathbf {y}})$

where ρ denotes the dipole distribution, ∂/∂ν denotes the directional derivative in the direction of the outward unit normal in the y variable, and dσ is the surface measure on S.

More generally, a double layer potential is associated to a hypersurface S in n-dimensional Euclidean space by means of

$u({\mathbf {x}})=\int _{S}\rho ({\mathbf {y}}){\frac {\partial }{\partial \nu }}P({\mathbf {x}}-{\mathbf {y}})\,d\sigma ({\mathbf {y}})$

where P(y) is the Newtonian kernel in n dimensions.

See also

• Single layer potential
• Potential theory
• Electrostatics
• Laplacian of the indicator

References

• Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume II, Wiley-Interscience.
• Kellogg, O. D. (1953), Foundations of potential theory, New York: Dover Publications, ISBN 978-0-486-60144-1.
• Shishmarev, I.A. (2001), "Double-layer potential", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
• Solomentsev, E.D. (2001), "Multi-pole potential", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.