Double-layer potential

An expression of the type

$$ \tag{1 } u ( x) = \int\limits _ \Gamma \frac \partial {\partial n _ {y} } ( h ( r _ {xy} )) \mu ( y) ds _ {y} , $$

where $ \Gamma $ is the boundary of an arbitrary bounded $ N $- dimensional domain $ G \subset \mathbf R ^ {N} $, $ N \geq 2 $, and $ n _ {y} $ is the exterior normal to the boundary $ \Gamma $ of $ G $ at a point $ y $; $ \mu $ is the potential density, which is a function defined on $ \Gamma $; $ h $ is a fundamental solution of the Laplace equation:

$$ \tag{2 } h ( r _ {xy} ) = \left \{ \begin{array}{ll} \frac{1}{( N - 2 ) \omega _ {N} } r _ {xy} ^ {2-} N , & N > 2 \\ \frac{1}{2 \pi } \mathop{\rm ln} \frac{1}{r} _ {xy} , & N = 2 , \\ \end{array} \right .$$

$ \omega _ {N} = 2 ( \sqrt \pi ) ^ {N} / \Gamma ( N / 2 ) $ is the area of the surface of the $ ( N - 1 ) $- dimensional unit sphere, and $ r _ {xy} = \sqrt {\sum _ {i=} 1 ^ {N} ( x _ {i} - y _ {i} ) ^ {2} } $ is the distance between two points $ x $ and $ y \in \mathbf R ^ {N} $. The boundary $ \Gamma $ is of class $ C ^ {( 1 , \lambda ) } $; it is a Lyapunov surface or a Lyapunov arc (cf. Lyapunov surfaces and curves).

Expression (1) may be interpreted as the potential produced by dipoles located on $ \Gamma $, the direction of which at any point $ y \in \Gamma $ coincides with that of the exterior normal $ n _ {y} $, while its intensity is equal to $ \mu ( y) $.

If $ \mu \in C ^ {(} 0) ( \Gamma ) $, then $ u $ is defined on $ \mathbf R ^ {N} $( in particular, on $ \Gamma $) and displays the following properties.

1) The function $ u $ has derivatives of all orders $ ( \in C ^ {( \infty ) } ) $ everywhere in $ \mathbf R ^ {N} \setminus \Gamma $ and satisfies the Laplace equation, and the derivatives with respect to the coordinates of a point may be computed by differentiation of the integrand.

2) On passing through the boundary $ \Gamma $ the function $ u $ undergoes a break. Let $ x _ {0} $ be an arbitrary point on $ \Gamma $; let $ u ^ {+} ( x _ {0} ) $ and $ u ^ {-} ( x _ {0} ) $ be the interior and exterior boundary values; then $ u ^ \pm ( x _ {0} ) $ exist and are equal to

$$ \tag{3 } u ^ \pm ( x _ {0} ) = \pm \frac{\mu ( x _ {0} ) }{2} + \int\limits _ \Gamma \frac \partial {\partial n _ {y} } ( h ( r _ {x _ {0} y } ) ) \mu ( y) ds _ {y} , $$

and the integral in formula (3) as a function of $ x _ {0} \in \Gamma $ belongs to $ C ^ {( 0 , \alpha ) } $ for any $ 0 \leq \alpha < 1 $; also, the function equal to $ u $ in $ G $ and to $ u ^ {+} $ on $ \Gamma $ is continuous on $ G \cup \Gamma $, while the function equal to $ u $ in $ \mathbf R ^ {N} \setminus ( G \cup \Gamma ) $ and equal to $ u ^ {-} $ on $ \Gamma $ is continuous in $ \mathbf R ^ {N} \setminus G $.

3) If the density $ \mu \in C ^ {( 0, \alpha ) } $ and if $ \alpha \leq \lambda $, then $ u $, extended as in (2) on $ G \cup \Gamma $ or $ \mathbf R ^ {N} \setminus G $, is of class $ C ^ {( 0, \alpha ) } $ in $ G \cup \Gamma $ or in $ \mathbf R ^ {N} \setminus G $.

4) If $ \alpha > 1 - \lambda $, and $ x _ {1} $ and $ x _ {2} $ are two points on the normal issuing from a point $ x _ {0} $ and lying symmetric about $ x _ {0} $, then

$$ \tag{4 } \lim\limits _ {x _ {1} \rightarrow x _ {0} } \left ( \frac{\partial u ( x _ {2} ) }{\partial n } - \frac{\partial u ( x _ {1} ) }{\partial n } \right ) = 0. $$

In particular, if one of the derivatives $ \partial u ^ {+} ( x _ {0} ) / \partial n $, $ \partial u ^ {-} ( x _ {0} ) / \partial n $ exists, then the other derivative also exists and $ \partial u ^ {+} ( x _ {0} ) / \partial n = \partial u ^ {-} ( x _ {0} ) / \partial n $. This is also true if $ \mu \in C ^ {(} 0) ( \Gamma ) $ and $ \Gamma \in C ^ {(} 2) $.

The above properties can be generalized in various ways. The density $ \mu $ may belong to $ L _ {p} ( \Gamma ) $, $ p \geq 1 $. Then $ u \in L _ {p} ( G \cup \Gamma ) $, $ u \in C ^ {( \infty ) } $ outside $ \Gamma $ and it satisfies the Laplace equation, formula (3) and (4) apply for almost-all $ x _ {0} \in \Gamma $ and the integral in (3) belongs to $ L _ {p} ( \Gamma ) $.

The properties of double-layer potentials, regarded as integrals with respect to an arbitrary measure $ \nu $ defined on $ \Gamma $, have also been studied:

$$ u ( x) = \int\limits _ \Gamma \frac \partial {\partial n _ {y} } ( h ( r _ {xy} ) ) d \nu ( y) . $$

Here, too, $ u \in C ^ {( \infty ) } $ outside $ \Gamma $ and it satisfies the Laplace equation. Formulas (3) and (4) apply for almost-all $ x _ {0} \in \Gamma $ with respect to the Lebesgue measure $ \nu $ after $ \mu $ has been replaced by the density $ \nu ^ \prime $. In definition (1) the fundamental solution of the Laplace equation may be replaced by an arbitrary Lewy function for a general elliptic operator of the second order with variable coefficients, while $ \partial / \partial n _ {y} $ is replaced by the derivative with respect to the conormal. The properties listed above remain valid [2].

The double-layer potential plays an important role in solving boundary value problems of elliptic equations. The representation of the solution of the (first) boundary value problem is sought as a double-layer potential with unknown density $ \mu $ and an application of property (2) leads to a Fredholm equation of the second kind on $ \Gamma $ in order to determine the function $ \mu $[1], [2]. In solving boundary value problems for parabolic equations use is made of the concept of the thermal double layer potential, i.e. of an integral of the type

$$ \nu ( x , t ) = \int\limits _ { 0 } ^ { t } d \tau \int\limits _ \Gamma \frac \partial {\partial n _ {y} } ( G ( x, t; y, \tau ) ) \sigma ( y, \tau ) dy , $$

where $ G ( x, t; y , \tau ) $ is a fundamental solution of the thermal conductance (or heat) equation in an $ N $- dimensional space:

$$ G ( x, t; y , \tau ) = \frac{1}{( 2 \sqrt \pi ) ^ {N} ( t - \tau ) ^ {N/2} } e ^ {- r _ {xy} ^ {2} / 4 ( t - \tau ) } . $$

Here, $ \sigma $ is the potential density. The function $ \nu $ and its generalization to the case of an arbitrary parabolic equation of the second order have properties which are similar to those described above for $ u $[3], [4], [5].

References

[1]	N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from French)
[2]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[3]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[4]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[5]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)

Comments

See [a1] for an introduction to double-layer potentials for more general open sets in $ \mathbf R ^ {n} $.

References

[a1]	J. Král, "Integral operators in potential theory" , Lect. notes in math. , 823 , Springer (1980)