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An expression of the type
 
An expression of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
u ( x)  = \int\limits _  \Gamma 
 +
\frac \partial {\partial  n _ {y} }
 +
 
 +
( h ( r _ {xy} )) \mu ( y) ds _ {y} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338802.png" /> is the boundary of an arbitrary bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338803.png" />-dimensional domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338805.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338806.png" /> is the exterior normal to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338807.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338808.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d0338809.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388010.png" /> is the potential density, which is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388011.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388012.png" /> is a fundamental solution of the Laplace equation:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
h ( r _ {xy} ) =  \left \{
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388014.png" /> is the area of the surface of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388015.png" />-dimensional unit sphere, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388016.png" /> is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388018.png" />. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388019.png" /> is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388020.png" />; it is a Lyapunov surface or a Lyapunov arc (cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]).
+
\begin{array}{ll}
  
Expression (1) may be interpreted as the potential produced by dipoles located on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388021.png" />, the direction of which at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388022.png" /> coincides with that of the exterior normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388023.png" />, while its intensity is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388024.png" />.
+
\frac{1}{( N - 2 ) \omega _ {N} }
 +
r _ {xy}  ^ {2-} N , & N > 2  \\
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388026.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388027.png" /> (in particular, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388028.png" />) and displays the following properties.
+
\frac{1}{2 \pi }
 +
  \mathop{\rm ln} 
 +
\frac{1}{r} _ {xy} , & N = 2 , \\
 +
\end{array}
  
1) The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388029.png" /> has derivatives of all orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388030.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388031.png" /> and satisfies the Laplace equation, and the derivatives with respect to the coordinates of a point may be computed by differentiation of the integrand.
+
\right .$$
  
2) On passing through the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388032.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388033.png" /> undergoes a break. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388034.png" /> be an arbitrary point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388035.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388037.png" /> be the interior and exterior boundary values; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388038.png" /> exist and are equal to
+
$  \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|Lyapunov surfaces and curves]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
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) $.
  
and the integral in formula (3) as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388040.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388041.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388042.png" />; also, the function equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388044.png" /> and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388046.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388047.png" />, while the function equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388048.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388049.png" /> and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388051.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388052.png" />.
+
If  $  \mu \in C  ^ {(} 0) ( \Gamma ) $,  
 +
then  $  u $
 +
is defined on $  \mathbf R  ^ {N} $(
 +
in particular, on  $  \Gamma $)
 +
and displays the following properties.
  
3) If the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388053.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388055.png" />, extended as in (2) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388056.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388057.png" />, is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388058.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388059.png" /> or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388060.png" />.
+
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.
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388061.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388063.png" /> are two points on the normal issuing from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388064.png" /> and lying symmetric about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388065.png" />, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{3 }
 +
u  ^  \pm  ( x _ {0} )  = \pm 
 +
\frac{\mu ( x _ {0} ) }{2}
 +
+
 +
\int\limits _  \Gamma 
 +
\frac \partial {\partial  n _ {y} }
  
In particular, if one of the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388068.png" /> exists, then the other derivative also exists and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388069.png" />. This is also true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388071.png" />.
+
( h ( r _ {x _ {0}  y } ) ) \mu ( y)  ds _ {y} ,
 +
$$
  
The above properties can be generalized in various ways. The density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388072.png" /> may belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388074.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388076.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388077.png" /> and it satisfies the Laplace equation, formula (3) and (4) apply for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388078.png" /> and the integral in (3) belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388079.png" />.
+
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 $.
  
The properties of double-layer potentials, regarded as integrals with respect to an arbitrary measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388080.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388081.png" />, have also been studied:
+
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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388082.png" /></td> </tr></table>
+
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
  
Here, too, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388083.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388084.png" /> and it satisfies the Laplace equation. Formulas (3) and (4) apply for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388085.png" /> with respect to the Lebesgue measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388086.png" /> after <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388087.png" /> has been replaced by the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388088.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388089.png" /> is replaced by the derivative with respect to the conormal. The properties listed above remain valid [[#References|[2]]].
+
$$ \tag{4 }
 +
\lim\limits _ {x _ {1} \rightarrow x _ {0} }
 +
\left (  
 +
\frac{\partial  u ( x _ {2} ) }{\partial  n }
 +
-
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388090.png" /> and an application of property (2) leads to a [[Fredholm equation|Fredholm equation]] of the second kind on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388091.png" /> in order to determine the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388092.png" /> [[#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
+
\frac{\partial  u ( x _ {1} ) }{\partial  n }
 +
\right ) = 0.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388093.png" /></td> </tr></table>
+
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) $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388094.png" /> is a fundamental solution of the thermal conductance (or heat) equation in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388095.png" />-dimensional space:
+
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 ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388096.png" /></td> </tr></table>
+
The properties of double-layer potentials, regarded as integrals with respect to an arbitrary measure  $  \nu $
 +
defined on  $  \Gamma $,
 +
have also been studied:
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388097.png" /> is the potential density. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388098.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d03388099.png" /> [[#References|[3]]], [[#References|[4]]], [[#References|[5]]].
+
$$
 +
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]]].
  
 
====References====
 
====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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====
See [[#References|[a1]]] for an introduction to double-layer potentials for more general open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033880/d033880100.png" />.
+
See [[#References|[a1]]] for an introduction to double-layer potentials for more general open sets in $  \mathbf R  ^ {n} $.
  
 
====References====
 
====References====
 
<table><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>
 
<table><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>

Revision as of 19:36, 5 June 2020


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)
How to Cite This Entry:
Double-layer potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double-layer_potential&oldid=46766
This article was adapted from an original article by I.A. Shishmarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article