Eckmann–Hilton duality

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

• Potential theory
• Electrostatics
• Laplacian of the indicator

References

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