View source for Double-layer potential

<!--
d0338801.png
$#A+1 = 100 n = 0
$#C+1 = 100 : ~/encyclopedia/old_files/data/D033/D.0303880 Double\AAhlayer potential
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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 [[#References|[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|Fredholm equation]] of the second kind on  $  \Gamma $
in order to determine the function  $  \mu $[[#References|[1]]], [[#References|[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 $[[#References|[3]]], [[#References|[4]]], [[#References|[5]]].

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

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. Günter,   "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from French)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  C. Miranda,   "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Král,   "Integral operators in potential theory" , ''Lect. notes in math.'' , '''823''' , Springer  (1980)</TD></TR>
</table>