Physics: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 $$u(\mathbf{𝐱})=\frac{-1}{4\pi }\underset{S}{\int }\rho (\mathbf{𝐲})\frac{\partial }{\partial \nu }\frac{1}{|\mathbf{𝐱}-\mathbf{𝐲}|}d\sigma (\mathbf{𝐲})$$ 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 $$u(\mathbf{𝐱})=\underset{S}{\int }\rho (\mathbf{𝐲})\frac{\partial }{\partial \nu }P(\mathbf{𝐱}-\mathbf{𝐲})d\sigma (\mathbf{𝐲})$$ 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 .
• Hazewinkel, Michiel, ed. (2001), "Double-layer potential", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=d/d033880 .
• Hazewinkel, Michiel, ed. (2001), "Multi-pole potential", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=m/m065210 .