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.

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, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/d/d033880.htm .
• Solomentsev, E.D. (2001), "Multi-pole potential", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/m/m065210.htm .