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 {{math|u(x)}} is a scalar-valued function of {{math|x ∈ R³}} given by

$$u(\backslash mathbf\{x\})\; =\; \backslash frac\; \{-1\}\; \{4\backslash pi\}\; \backslash int\_S\; \backslash rho(\backslash mathbf\{y\})\; \backslash frac\{\backslash partial\}\{\backslash partial\backslash nu\}\; \backslash frac\{1\}$$

\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

$$u(\backslash mathbf\{x\})\; =\; \backslash int\_S\; \backslash rho(\backslash mathbf\{y\})\backslash frac\{\backslash partial\}\{\backslash partial\backslash nu\}\; P(\backslash mathbf\{x\}-\backslash mathbf\{y\})\backslash ,d\backslash sigma(\backslash mathbf\{y\})$$

where P(y) is the Newtonian kernel in n dimensions.