Double layer potential explained

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 is a scalar-valued function of given by$$u(\backslash mathbf)\; =\; \backslash frac\; \backslash int\_S\; \backslash rho(\backslash mathbf)\; \backslash frac\; \backslash frac$$

\, d\sigma(\mathbf)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$$u(\backslash mathbf)\; =\; \backslash int\_S\; \backslash rho(\backslash mathbf)\backslash frac\; P(\backslash mathbf-\backslash mathbf)\backslash ,d\backslash sigma(\backslash mathbf)$$where P(y) is the Newtonian kernel in n dimensions.

See also

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

References

• .
• .
• .
• .