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 ) = − 1 4 π ∫ S ρ ( y ) ∂ ∂ ν 1 | x − y | d σ ( y ) {\displaystyle u(\mathbf {x} )={\frac {-1}{4\pi }}\int _{S}\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 ( x ) = ∫ S ρ ( y ) ∂ ∂ ν P ( x − y ) d σ ( y ) {\displaystyle u(\mathbf {x} )=\int _{S}\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

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.
  • Shishmarev, I.A. (2001) [1994], "Double-layer potential", Encyclopedia of Mathematics, EMS Press.
  • Solomentsev, E.D. (2001) [1994], "Multi-pole potential", Encyclopedia of Mathematics, EMS Press.