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Double layer potential

From Wikipedia, the free encyclopedia

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 xR3 given by 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 where P(y) is the Newtonian kernel in n dimensions.

See also

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References

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  • 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.