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 $$u(𝐱)=\frac{-1}{4\pi }{\int }_S\rho (𝐲)\frac{\partial }{\partial \nu }\frac{1}{|𝐱-𝐲|}d\sigma (𝐲)$$ 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 $$u(𝐱)={\int }_S\rho (𝐲)\frac{\partial }{\partial \nu }P(𝐱-𝐲)d\sigma (𝐲)$$ where P(y) is the Newtonian kernel in n dimensions.

See also

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

References

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