Double-layer potential
An expression of the type
 | (1) |
where 
is the boundary of an arbitrary bounded 
-dimensional domain 
, 
, and 
is the exterior normal to the boundary 
of 
at a point 
; 
is the potential density, which is a function defined on 
; 
is a fundamental solution of the Laplace equation:
 | (2) |
is the area of the surface of the 
-dimensional unit sphere, and 
is the distance between two points 
and 
. The boundary 
is of class 
; 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 
, the direction of which at any point 
coincides with that of the exterior normal 
, while its intensity is equal to 
.
If 
, then 
is defined on 
(in particular, on 
) and displays the following properties.
1) The function 
has derivatives of all orders 
everywhere in 
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 
the function 
undergoes a break. Let 
be an arbitrary point on 
; let 
and 
be the interior and exterior boundary values; then 
exist and are equal to
 | (3) |
and the integral in formula (3) as a function of 
belongs to 
for any 
; also, the function equal to 
in 
and to 
on 
is continuous on 
, while the function equal to 
in 
and equal to 
on 
is continuous in 
.
3) If the density 
and if 
, then 
, extended as in (2) on 
or 
, is of class 
in 
or in 
.
4) If 
, and 
and 
are two points on the normal issuing from a point 
and lying symmetric about 
, then
 | (4) |
In particular, if one of the derivatives 
, 
exists, then the other derivative also exists and 
. This is also true if 
and 
.
The above properties can be generalized in various ways. The density 
may belong to 
, 
. Then 
, 
outside 
and it satisfies the Laplace equation, formula (3) and (4) apply for almost-all 
and the integral in (3) belongs to 
.
The properties of double-layer potentials, regarded as integrals with respect to an arbitrary measure 
defined on 
, have also been studied:
 |
Here, too, 
outside 
and it satisfies the Laplace equation. Formulas (3) and (4) apply for almost-all 
with respect to the Lebesgue measure 
after 
has been replaced by the density 
. 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 
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 
and an application of property (2) leads to a Fredholm equation of the second kind on 
in order to determine the function 
[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
 |
where 
is a fundamental solution of the thermal conductance (or heat) equation in an 
-dimensional space:
 |
Here, 
is the potential density. The function 
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 
[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 
.
References
| [a1] | J. Král, "Integral operators in potential theory" , Lect. notes in math. , 823 , Springer (1980) |
Double-layer potential. I.A. Shishmarev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Double-layer_potential&oldid=16189