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An expression
 
An expression
  
<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/s/s085/s085260/s0852601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
u ( x)  = \int\limits _ { S }
 +
h ( | x - y | ) f ( y) \
 +
d \sigma ( y) ,
 +
$$
 +
 
 +
where  $  S $
 +
is a closed Lyapunov surface (of class  $  C ^ {1 , \lambda } $,
 +
cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]) in the Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,
 +
separating  $  \mathbf R  ^ {n} $
 +
into an interior domain  $  D  ^ {+} $
 +
and an exterior domain  $  D  ^ {-} $;  
 +
$  h ( | x - y | ) $
 +
is a [[Fundamental solution|fundamental solution]] of the [[Laplace operator|Laplace operator]]:
 +
 
 +
$$
 +
h ( | x - y | )  = \
 +
\left \{
 +
\begin{array}{cc}
 +
 
 +
\frac{1}{( n - 2 ) \omega _ {n} | x - y | ^ {n - 2 } }
 +
,  & n \geq  3 ;  \\
 +
 
 +
\frac{1}{2 \pi }
 +
\
 +
\mathop{\rm ln} 
 +
\frac{1}{| x - y | }
 +
,  & n = 2 ;  \\
 +
\end{array}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852602.png" /> is a closed Lyapunov surface (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852603.png" />, cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]) in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852605.png" />, separating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852606.png" /> into an interior domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852607.png" /> and an exterior domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852608.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s0852609.png" /> is a [[Fundamental solution|fundamental solution]] of the [[Laplace operator|Laplace operator]]:
+
\right .$$
  
<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/s/s085/s085260/s08526010.png" /></td> </tr></table>
+
$  \omega _ {n} = 2 \pi  ^ {n/2} / \Gamma ( n / 2 ) $
 +
is the area of the unit sphere in  $  \mathbf R  ^ {n} $;  
 +
$  | x - y | $
 +
is the distance between two points  $  x $
 +
and  $  y $;  
 +
and  $  d \sigma ( y) $
 +
is the area element on  $  S $.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526011.png" /> is the area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526013.png" /> is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526015.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526016.png" /> is the area element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526017.png" />.
+
If  $  f \in C  ^ {(} 0) ( S) $,
 +
then  $  u $
 +
is everywhere defined on  $  \mathbf R  ^ {n} $.  
 +
A simple-layer potential is a particular case of a [[Newton potential|Newton potential]], generated by masses distributed on  $  S $
 +
with surface density  $  f $,
 +
and with the following properties.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526019.png" /> is everywhere defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526020.png" />. A simple-layer potential is a particular case of a [[Newton potential|Newton potential]], generated by masses distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526021.png" /> with surface density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526022.png" />, and with the following properties.
+
In  $  D  ^ {+} $
 +
and  $  D  ^ {-} $
 +
a simple-layer potential  $  u $
 +
has derivatives of all orders, which can be calculated by differentiation under the integral sign, and satisfies the [[Laplace equation|Laplace equation]],  $  \Delta u = 0 $,  
 +
i.e. it is a [[Harmonic function|harmonic function]]. For  $  n \geq  3 $
 +
it is a function regular at infinity,  $  u ( \infty ) = 0 $.  
 +
A simple-layer potential is continuous throughout  $  \mathbf R  ^ {n} $,
 +
and  $  u \in C ^ {( 0 , \nu ) } ( \mathbf R  ^ {n} ) $
 +
for any  $  \nu $,  
 +
$  0 < \nu < \lambda $.  
 +
When passing through the surface $  S $,
 +
the derivative along the outward normal  $  \mathbf n _ {0} $
 +
to  $  S $
 +
at a point  $  y _ {0} \in S $
 +
undergoes a discontinuity. The limit values of the normal derivative from  $  D  ^ {+} $
 +
and  $  D  ^ {-} $
 +
exist, are everywhere continuous on  $  S $,  
 +
and can be expressed, respectively, by the formula:
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526024.png" /> a simple-layer potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526025.png" /> has derivatives of all orders, which can be calculated by differentiation under the integral sign, and satisfies the [[Laplace equation|Laplace equation]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526026.png" />, i.e. it is a [[Harmonic function|harmonic function]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526027.png" /> it is a function regular at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526028.png" />. A simple-layer potential is continuous throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526029.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526030.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526032.png" />. When passing through the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526033.png" />, the derivative along the outward normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526034.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526035.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526036.png" /> undergoes a discontinuity. The limit values of the normal derivative from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526038.png" /> exist, are everywhere continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526039.png" />, and can be expressed, respectively, by the formula:
+
$$ \tag{2 }
 +
\left .
 +
\lim\limits _ {x \rightarrow y _ {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/s/s085/s085260/s08526040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{du}{d \mathbf n _ {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/s/s085/s085260/s08526041.png" /></td> </tr></table>
+
\right | _ {i}  = \
 +
 
 +
\frac{d u ( y _ {0} ) }{d \mathbf n _ {0} }
 +
-
 +
 
 +
\frac{f ( y _ {0} ) }{2}
 +
,\  x \in D  ^ {+} ,
 +
$$
 +
 
 +
$$
 +
\left . \lim\limits _ {x \rightarrow y _ {0} } 
 +
\frac{du}{d
 +
\mathbf n _ {0} }
 +
\right | _ {e}  =
 +
\frac{d u ( y _ {0} ) }{d \mathbf n _ {0} }
 +
+
 +
\frac{f ( y _ {0} ) }{2}
 +
,\  x \in D  ^ {-} ,
 +
$$
  
 
where
 
where
  
<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/s/s085/s085260/s08526042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
  
is the so-called direct value of the normal derivative of a simple-layer potential at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526043.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526044.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526046.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526047.png" />, then the partial derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526048.png" /> can be continuously extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526050.png" /> as functions of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526052.png" />, respectively. In this case one also has
+
\frac{d u ( y _ {0} ) }{d \mathbf n _ {0} }
 +
  = \
 +
\int\limits _ { S }
  
<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/s/s085/s085260/s08526053.png" /></td> </tr></table>
+
\frac \partial {\partial  \mathbf n _ {0} }
  
These properties can be generalized in various directions. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526055.png" /> inside and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526056.png" />, formulas (2) hold almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526057.png" />, and the integral in (3) is summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526058.png" />. One has also studied properties of simple-layer potentials understood as integrals with respect to arbitrary Radon measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526059.png" /> concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526060.png" />:
+
h ( | y - y _ {0} | ) f ( y) d \sigma ( y )
 +
$$
  
<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/s/s085/s085260/s08526061.png" /></td> </tr></table>
+
is the so-called direct value of the normal derivative of a simple-layer potential at a point  $  y _ {0} \in S $.
 +
Moreover,  $  ( d u / d \mathbf n _ {0} ) ( y _ {0} ) \in C ^ {( 0 , \nu ) } ( S) $
 +
for all  $  \nu $,
 +
$  0 < \nu < \lambda $.
 +
If  $  f( y) \in C ^ {( 0 , \nu ) } ( S) $,
 +
then the partial derivatives of  $  u( x ) $
 +
can be continuously extended to  $  \overline{ {D  ^ {+} }}\; $
 +
and  $  \overline{ {D  ^ {-} }}\; $
 +
as functions of the classes  $  C ^ {( 0 , \nu ) } ( \overline{ {D  ^ {+} }}\; ) $
 +
and  $  C ^ {( 0 , \nu ) } ( \overline{ {D  ^ {-} }}\; ) $,
 +
respectively. In this case one also has
  
Here, also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526062.png" /> is a harmonic function outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526063.png" />, and formulas (2) hold almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526064.png" /> with respect to the Lebesgue measure, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526065.png" /> is replaced by the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526066.png" /> of the measure. In definition (1) one can replace the fundamental solution of the Laplace equation by an arbitrary Lewy function of a general second-order elliptic operator with variable coefficients of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526067.png" />, replacing the normal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526068.png" /> by the derivative along the co-normal. The properties listed remain true in this case (cf. [[#References|[2]]], [[#References|[3]]], [[#References|[4]]]).
+
$$
  
Simple-layer potentials are used in solving boundary value problems for elliptic equations. The solution of a second boundary value problem with prescribed normal derivative is represented as a simple-layer potential with unknown density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526069.png" />; the use of (2) and (3) leads to a Fredholm integral equation of the second kind on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526070.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526071.png" /> (cf. [[#References|[2]]]–[[#References|[5]]]).
+
\frac{d u }{d \mathbf n _ {0} }
 +
( y _ {0} )
 +
\in  C ^ {( 0 , \lambda ) } ( S ) .
 +
$$
 +
 
 +
These properties can be generalized in various directions. E.g., if  $  f \in L _ {1} ( S) $,
 +
then  $  u \in L _ {1} $
 +
inside and on  $  S $,
 +
formulas (2) hold almost everywhere on  $  S $,
 +
and the integral in (3) is summable on  $  S $.
 +
One has also studied properties of simple-layer potentials understood as integrals with respect to arbitrary Radon measures  $  \mu $
 +
concentrated on  $  S $:
 +
 
 +
$$
 +
u ( x)  =  \int\limits
 +
h ( | x - y | ) \
 +
d \mu ( y ) .
 +
$$
 +
 
 +
Here, also,  $  u $
 +
is a harmonic function outside  $  S $,
 +
and formulas (2) hold almost everywhere on  $  S $
 +
with respect to the Lebesgue measure, where  $  f ( y _ {0} ) $
 +
is replaced by the derivative  $  \mu  ^  \prime  ( y _ {0} ) $
 +
of the measure. In definition (1) one can replace the fundamental solution of the Laplace equation by an arbitrary Lewy function of a general second-order elliptic operator with variable coefficients of class  $  C ^ {( 0 , \lambda ) } $,
 +
replacing the normal derivative  $  d / d \mathbf n _ {0} $
 +
by the derivative along the co-normal. The properties listed remain true in this case (cf. [[#References|[2]]], [[#References|[3]]], [[#References|[4]]]).
 +
 
 +
Simple-layer potentials are used in solving boundary value problems for elliptic equations. The solution of a second boundary value problem with prescribed normal derivative is represented as a simple-layer potential with unknown density $  f $;  
 +
the use of (2) and (3) leads to a Fredholm integral equation of the second kind on $  S $
 +
for $  f $(
 +
cf. [[#References|[2]]]–[[#References|[5]]]).
  
 
In solving boundary value problems for parabolic equations one uses simple-layer heat potentials, of the form
 
In solving boundary value problems for parabolic equations one uses simple-layer heat potentials, of the form
  
<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/s/s085/s085260/s08526072.png" /></td> </tr></table>
+
$$
 +
v ( x , t )  = \
 +
\int\limits _ { 0 } ^ { t }  \int\limits _ { S }
 +
G ( x , t ; y , \tau )
 +
f ( y , \tau )  d \sigma ( y)  d \tau ,
 +
$$
  
 
where
 
where
  
<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/s/s085/s085260/s08526073.png" /></td> </tr></table>
+
$$
 +
G ( x , t ; y , \tau )  = \
  
is the fundamental solution of the heat equation in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526074.png" />-dimensional space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526075.png" /> is the density. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526076.png" /> and its generalization to arbitrary second-order parabolic equations have properties analogous to those indicated for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526077.png" /> (cf. [[#References|[3]]], [[#References|[4]]], [[#References|[6]]]).
+
\frac{1}{( 2 \sqrt \pi )  ^ {n} ( t - \tau )  ^ {n/2} }
 +
 
 +
\mathop{\rm exp} \
 +
\left [
 +
 
 +
\frac{- | x - y |  ^ {2} }{4 ( t - \tau ) }
 +
 
 +
\right ]
 +
$$
 +
 
 +
is the fundamental solution of the heat equation in the $  n $-
 +
dimensional space, and $  f ( y , \tau ) $
 +
is the density. The function $  v $
 +
and its generalization to arbitrary second-order parabolic equations have properties analogous to those indicated for $  u $(
 +
cf. [[#References|[3]]], [[#References|[4]]], [[#References|[6]]]).
  
 
====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 Russian)</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. Tikhonov,  A.A. Samarskii,  "Equations of mathematical physics" , Pergamon  (1963)  (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">[6]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</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 Russian)</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. Tikhonov,  A.A. Samarskii,  "Equations of mathematical physics" , Pergamon  (1963)  (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">[6]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
See [[#References|[a1]]] for simple-layer potentials on more general open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085260/s08526078.png" />.
+
See [[#References|[a1]]] for simple-layer potentials on 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>

Latest revision as of 14:55, 7 June 2020


An expression

$$ \tag{1 } u ( x) = \int\limits _ { S } h ( | x - y | ) f ( y) \ d \sigma ( y) , $$

where $ S $ is a closed Lyapunov surface (of class $ C ^ {1 , \lambda } $, cf. Lyapunov surfaces and curves) in the Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, separating $ \mathbf R ^ {n} $ into an interior domain $ D ^ {+} $ and an exterior domain $ D ^ {-} $; $ h ( | x - y | ) $ is a fundamental solution of the Laplace operator:

$$ h ( | x - y | ) = \ \left \{ \begin{array}{cc} \frac{1}{( n - 2 ) \omega _ {n} | x - y | ^ {n - 2 } } , & n \geq 3 ; \\ \frac{1}{2 \pi } \ \mathop{\rm ln} \frac{1}{| x - y | } , & n = 2 ; \\ \end{array} \right .$$

$ \omega _ {n} = 2 \pi ^ {n/2} / \Gamma ( n / 2 ) $ is the area of the unit sphere in $ \mathbf R ^ {n} $; $ | x - y | $ is the distance between two points $ x $ and $ y $; and $ d \sigma ( y) $ is the area element on $ S $.

If $ f \in C ^ {(} 0) ( S) $, then $ u $ is everywhere defined on $ \mathbf R ^ {n} $. A simple-layer potential is a particular case of a Newton potential, generated by masses distributed on $ S $ with surface density $ f $, and with the following properties.

In $ D ^ {+} $ and $ D ^ {-} $ a simple-layer potential $ u $ has derivatives of all orders, which can be calculated by differentiation under the integral sign, and satisfies the Laplace equation, $ \Delta u = 0 $, i.e. it is a harmonic function. For $ n \geq 3 $ it is a function regular at infinity, $ u ( \infty ) = 0 $. A simple-layer potential is continuous throughout $ \mathbf R ^ {n} $, and $ u \in C ^ {( 0 , \nu ) } ( \mathbf R ^ {n} ) $ for any $ \nu $, $ 0 < \nu < \lambda $. When passing through the surface $ S $, the derivative along the outward normal $ \mathbf n _ {0} $ to $ S $ at a point $ y _ {0} \in S $ undergoes a discontinuity. The limit values of the normal derivative from $ D ^ {+} $ and $ D ^ {-} $ exist, are everywhere continuous on $ S $, and can be expressed, respectively, by the formula:

$$ \tag{2 } \left . \lim\limits _ {x \rightarrow y _ {0} } \ \frac{du}{d \mathbf n _ {0} } \right | _ {i} = \ \frac{d u ( y _ {0} ) }{d \mathbf n _ {0} } - \frac{f ( y _ {0} ) }{2} ,\ x \in D ^ {+} , $$

$$ \left . \lim\limits _ {x \rightarrow y _ {0} } \frac{du}{d \mathbf n _ {0} } \right | _ {e} = \frac{d u ( y _ {0} ) }{d \mathbf n _ {0} } + \frac{f ( y _ {0} ) }{2} ,\ x \in D ^ {-} , $$

where

$$ \tag{3 } \frac{d u ( y _ {0} ) }{d \mathbf n _ {0} } = \ \int\limits _ { S } \frac \partial {\partial \mathbf n _ {0} } h ( | y - y _ {0} | ) f ( y) d \sigma ( y ) $$

is the so-called direct value of the normal derivative of a simple-layer potential at a point $ y _ {0} \in S $. Moreover, $ ( d u / d \mathbf n _ {0} ) ( y _ {0} ) \in C ^ {( 0 , \nu ) } ( S) $ for all $ \nu $, $ 0 < \nu < \lambda $. If $ f( y) \in C ^ {( 0 , \nu ) } ( S) $, then the partial derivatives of $ u( x ) $ can be continuously extended to $ \overline{ {D ^ {+} }}\; $ and $ \overline{ {D ^ {-} }}\; $ as functions of the classes $ C ^ {( 0 , \nu ) } ( \overline{ {D ^ {+} }}\; ) $ and $ C ^ {( 0 , \nu ) } ( \overline{ {D ^ {-} }}\; ) $, respectively. In this case one also has

$$ \frac{d u }{d \mathbf n _ {0} } ( y _ {0} ) \in C ^ {( 0 , \lambda ) } ( S ) . $$

These properties can be generalized in various directions. E.g., if $ f \in L _ {1} ( S) $, then $ u \in L _ {1} $ inside and on $ S $, formulas (2) hold almost everywhere on $ S $, and the integral in (3) is summable on $ S $. One has also studied properties of simple-layer potentials understood as integrals with respect to arbitrary Radon measures $ \mu $ concentrated on $ S $:

$$ u ( x) = \int\limits h ( | x - y | ) \ d \mu ( y ) . $$

Here, also, $ u $ is a harmonic function outside $ S $, and formulas (2) hold almost everywhere on $ S $ with respect to the Lebesgue measure, where $ f ( y _ {0} ) $ is replaced by the derivative $ \mu ^ \prime ( y _ {0} ) $ of the measure. In definition (1) one can replace the fundamental solution of the Laplace equation by an arbitrary Lewy function of a general second-order elliptic operator with variable coefficients of class $ C ^ {( 0 , \lambda ) } $, replacing the normal derivative $ d / d \mathbf n _ {0} $ by the derivative along the co-normal. The properties listed remain true in this case (cf. [2], [3], [4]).

Simple-layer potentials are used in solving boundary value problems for elliptic equations. The solution of a second boundary value problem with prescribed normal derivative is represented as a simple-layer potential with unknown density $ f $; the use of (2) and (3) leads to a Fredholm integral equation of the second kind on $ S $ for $ f $( cf. [2][5]).

In solving boundary value problems for parabolic equations one uses simple-layer heat potentials, of the form

$$ v ( x , t ) = \ \int\limits _ { 0 } ^ { t } \int\limits _ { S } G ( x , t ; y , \tau ) f ( y , \tau ) d \sigma ( y) d \tau , $$

where

$$ G ( x , t ; y , \tau ) = \ \frac{1}{( 2 \sqrt \pi ) ^ {n} ( t - \tau ) ^ {n/2} } \mathop{\rm exp} \ \left [ \frac{- | x - y | ^ {2} }{4 ( t - \tau ) } \right ] $$

is the fundamental solution of the heat equation in the $ n $- dimensional space, and $ f ( y , \tau ) $ is the density. The function $ v $ and its generalization to arbitrary second-order parabolic equations have properties analogous to those indicated for $ u $( cf. [3], [4], [6]).

References

[1] N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian)
[2] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[3] A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (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)
[6] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)

Comments

See [a1] for simple-layer potentials on 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:
Simple-layer potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple-layer_potential&oldid=49425
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article