# 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)

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