View source for Simple-layer potential

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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|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}

 \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|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|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:

$$ \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. [[#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

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

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. [[#References|[3]]], [[#References|[4]]], [[#References|[6]]]).

====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>

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

====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>