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A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464701.png" />, defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464702.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464704.png" />, having continuous partial derivatives of the first and second orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464705.png" />, and which is a solution of the [[Laplace equation|Laplace equation]]
+
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$#A+1 = 237 n = 0
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$#C+1 = 237 : ~/encyclopedia/old_files/data/H046/H.0406470 Harmonic function
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<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/h/h046/h046470/h0464706.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464707.png" /> are the orthogonal Cartesian coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464708.png" />. This definition is sometimes extended to include complex functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h0464709.png" /> as well, in the sense that their real and imaginary parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647011.png" /> are harmonic functions. The requirements of continuity and even of the existence of derivatives are not a priori indispensable. For instance, one of Privalov's theorems is applicable: A continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647013.png" /> is a harmonic function if and only if at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647014.png" /> the mean-value property
+
A real-valued function  $  u $,  
 +
defined in a domain  $  D $
 +
of a Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,  
 +
having continuous partial derivatives of the first and second orders in $  D $,
 +
and which is a solution of the [[Laplace equation|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/h/h046/h046470/h04647015.png" /></td> </tr></table>
+
$$
 +
\Delta u  \equiv \
  
— where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647016.png" /> is the ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647017.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647019.png" /> is the volume of this ball and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647020.png" /> is the volume element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647021.png" /> — is fulfilled for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647022.png" />.
+
\frac{\partial  ^ {2} u }{\partial  x _ {1}  ^ {2} }
 +
+ \dots +
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647023.png" /> is unbounded with a compact boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647024.png" />, the definition of a harmonic function may be completed to include the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647025.png" />, i.e. it may additionally be defined in domains in the Aleksandrov compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647026.png" />. The general principle of such a completion of the definition is that, under the simplest transformations which preserve harmonicity ([[Inversion|inversion]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647027.png" />, Kelvin transformations if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647028.png" />, cf. [[Kelvin transformation|Kelvin transformation]]) and map a finite point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647029.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647030.png" />, a harmonic function in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647031.png" /> becomes a harmonic function in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647032.png" />. On this basis, a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647033.png" /> is said to be regular at infinity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647034.png" /> if
+
\frac{\partial  ^ {2} u }{\partial  x _ {n}  ^ {2} }
 +
  = 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/h/h046/h046470/h04647035.png" /></td> </tr></table>
+
where  $  x _ {1} \dots x _ {n} $
 +
are the orthogonal Cartesian coordinates of the point  $  x $.
 +
This definition is sometimes extended to include complex functions  $  w( x) = u( x) + iv( x) $
 +
as well, in the sense that their real and imaginary parts  $  \mathop{\rm Re}  w( x) = u ( x) $
 +
and  $  \mathop{\rm Im}  w ( x) = v ( x) $
 +
are harmonic functions. The requirements of continuity and even of the existence of derivatives are not a priori indispensable. For instance, one of Privalov's theorems is applicable: A continuous function  $  u $
 +
in  $  D $
 +
is a harmonic function if and only if at any point  $  x \in D $
 +
the mean-value property
  
Thus, for a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647036.png" /> which is regular at infinity one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647039.png" />, the condition
+
$$
 +
u ( x)  = \
 +
{
 +
\frac{1}{\omega _ {n} ( R) }
 +
}
 +
\int\limits _ {B _ {n} ( x, R) }
 +
u ( y)  dy
 +
$$
  
<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/h/h046/h046470/h04647040.png" /></td> </tr></table>
+
— where  $  B _ {n} ( x, R) $
 +
is the ball of radius  $  R $
 +
with centre at  $  x $,
 +
$  \omega _ {n} ( R) $
 +
is the volume of this ball and  $  dy $
 +
is the volume element in  $  \mathbf R  ^ {n} $—
 +
is fulfilled for sufficiently small  $  R > 0 $.
 +
 
 +
If  $  D $
 +
is unbounded with a compact boundary  $  \partial  D $,
 +
the definition of a harmonic function may be completed to include the point at infinity  $  \infty $,
 +
i.e. it may additionally be defined in domains in the Aleksandrov compactification of  $  \mathbf R  ^ {n} $.
 +
The general principle of such a completion of the definition is that, under the simplest transformations which preserve harmonicity ([[Inversion|inversion]] if  $  n = 2 $,
 +
Kelvin transformations if  $  n \geq  3 $,
 +
cf. [[Kelvin transformation|Kelvin transformation]]) and map a finite point  $  x _ {0} $
 +
into  $  \infty $,
 +
a harmonic function in a neighbourhood of  $  x _ {0} $
 +
becomes a harmonic function in a neighbourhood of  $  \infty $.
 +
On this basis, a harmonic function  $  u $
 +
is said to be regular at infinity for  $  n \geq  3 $
 +
if
 +
 
 +
$$
 +
\lim\limits _ {| x | \rightarrow \infty } \
 +
u ( x)  = 0,\ \
 +
| x | = \sqrt {x _ {1}  ^ {2} + \dots + x _ {n}  ^ {2} } .
 +
$$
 +
 
 +
Thus, for a harmonic function  $  u $
 +
which is regular at infinity one always has  $  u( \infty ) = 0 $
 +
if  $  n \geq  3 $.
 +
If  $  n = 2 $,
 +
the condition
 +
 
 +
$$
 +
u ( x)  =  O( 1),\  | x | \rightarrow \infty ,
 +
$$
  
 
which implies the existence of a finite limit
 
which implies the existence of a finite limit
  
<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/h/h046/h046470/h04647041.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {| x | \rightarrow \infty } \
 +
u ( x)  = u ( \infty ),
 +
$$
  
 
must be met. Harmonic functions in unbounded domains are usually understood to mean harmonic functions regular at infinity.
 
must be met. Harmonic functions in unbounded domains are usually understood to mean harmonic functions regular at infinity.
Line 25: Line 96:
 
In the theory of harmonic functions an important role is played by the principal fundamental solutions of the Laplace equation:
 
In the theory of harmonic functions an important role is played by the principal fundamental solutions 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/h/h046/h046470/h04647042.png" /></td> </tr></table>
+
$$
 +
h _ {2} ( x)  = \
 +
{
 +
\frac{1}{2 \pi }
 +
} \
 +
\mathop{\rm ln}  {
 +
\frac{1}{| x | }
 +
} \ \
 +
\textrm{ if } \
 +
n = 2,
 +
$$
 +
 
 +
$$
 +
h _ {n} ( x)  = {
 +
\frac{1}{( n - 2) \sigma _ {n} }
 +
}
 +
{
 +
\frac{1}{| x | ^ {n - 2 } }
 +
} \  \textrm{ if }  n \geq  3,
 +
$$
 +
 
 +
where  $  \sigma _ {n} $
 +
is the surface area of the unit sphere in  $  \mathbf R  ^ {n} $.
 +
If  $  | x | > 0 $,
 +
this is a harmonic function. The fundamental solutions may be used to write down the basic formula of the theory of harmonic functions, which expresses the values of a harmonic function  $  u ( x) $
 +
inside a domain  $  D $
 +
in terms of its values  $  u( y) $
 +
on the boundary  $  S = \partial  D $
 +
and the values of its derivative in the direction of the exterior normal  $  \partial  u ( y) / \partial  \nu $
 +
towards  $  S $
 +
at the point  $  y $:
 +
 
 +
$$
 +
\int\limits _ { S }  \left [
 +
h _ {n} ( x - y)
 +
 
 +
\frac{\partial  u ( y) }{\partial  \nu }
 +
 
 +
- u ( y)
 +
 
 +
\frac{\partial  h _ {n} ( x - y) }{\partial  \nu _ {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/h/h046/h046470/h04647043.png" /></td> </tr></table>
+
\right ]  d \sigma ( y) =
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647044.png" /> is the surface area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647045.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647046.png" />, this is a harmonic function. The fundamental solutions may be used to write down the basic formula of the theory of harmonic functions, which expresses the values of a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647047.png" /> inside a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647048.png" /> in terms of its values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647049.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647050.png" /> and the values of its derivative in the direction of the exterior normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647051.png" /> towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647052.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647053.png" />:
+
$$
 +
= \
 +
\left \{
 +
\begin{array}{ll}
 +
u ( x),  & x \in D,  \\
 +
0, & x \notin \overline{D}\; . \\
 +
\end{array}
  
<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/h/h046/h046470/h04647054.png" /></td> </tr></table>
+
\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/h/h046/h046470/h04647055.png" /></td> </tr></table>
+
This Green formula is valid, for example, if the function  $  u $
 +
and its partial derivatives of the first order are continuous in the closed domain  $  \overline{D}\; $,
 +
i.e. if  $  u \in C  ^ {1} ( \overline{D}\; ) $,
 +
the boundary  $  S $
 +
of which is a piecewise-smooth closed surface or curve. It yields a representation of an arbitrary harmonic function  $  u $
 +
as the sum of single- and double-layer potentials (cf. [[Potential theory|Potential theory]]). The densities of these potentials, i.e. the boundary values  $  \partial  u ( y) / \partial  \nu $
 +
and  $  u( y) $
 +
respectively, cannot be specified arbitrarily. There is an integral relationship between the two, in that the left-hand side of the last-named formula — the Green integral — must vanish for all points  $  x $
 +
outside  $  \overline{D}\; $.
 +
The basic formula of the theory of harmonic functions is a direct analogue of the fundamental formula of the theory of analytic functions — the integral formula of Cauchy (cf. [[Cauchy integral|Cauchy integral]]). This formula also remains valid if the principal fundamental solution  $  h _ {n} $
 +
in it is replaced by any other fundamental solution of the Laplace equation which is sufficiently smooth in  $  \overline{D}\; $,
 +
e.g. belongs to  $  C  ^ {1} ( \overline{D}\; ) $.
  
This Green formula is valid, for example, if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647056.png" /> and its partial derivatives of the first order are continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647057.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647058.png" />, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647059.png" /> of which is a piecewise-smooth closed surface or curve. It yields a representation of an arbitrary harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647060.png" /> as the sum of single- and double-layer potentials (cf. [[Potential theory|Potential theory]]). The densities of these potentials, i.e. the boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647062.png" /> respectively, cannot be specified arbitrarily. There is an integral relationship between the two, in that the left-hand side of the last-named formula — the Green integral — must vanish for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647063.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647064.png" />. The basic formula of the theory of harmonic functions is a direct analogue of the fundamental formula of the theory of analytic functions — the integral formula of Cauchy (cf. [[Cauchy integral|Cauchy integral]]). This formula also remains valid if the principal fundamental solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647065.png" /> in it is replaced by any other fundamental solution of the Laplace equation which is sufficiently smooth in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647066.png" />, e.g. belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647067.png" />.
+
The fundamental properties of harmonic functions, on the assumption that the boundary  $  S $
 +
of the domain $  D $
 +
is piecewise smooth, are listed below. After suitable modification, many of them are also valid for complex harmonic functions.
  
The fundamental properties of harmonic functions, on the assumption that the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647068.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647069.png" /> is piecewise smooth, are listed below. After suitable modification, many of them are also valid for complex harmonic functions.
+
1) If  $  D $
 +
is a bounded domain and a harmonic function  $  u \in C  ^ {1} ( \overline{D}\; ) $,  
 +
then
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647070.png" /> is a bounded domain and a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647071.png" />, then
+
$$
 +
\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/h/h046/h046470/h04647072.png" /></td> </tr></table>
+
\frac{\partial  u ( y) }{\partial  \nu }
 +
\
 +
d \sigma ( y)  = 0.
 +
$$
  
2) The mean-value theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647073.png" /> is a harmonic function in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647074.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647075.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647076.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647077.png" />, then its value at the centre of the ball is equal to the value of its arithmetical mean on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647078.png" />, i.e.
+
2) The mean-value theorem: If $  u $
 +
is a harmonic function in the ball $  B = B ( x _ {0,\ } R) $
 +
of radius $  R $
 +
with centre at $  x _ {0} $
 +
and if $  u \in C  ^ {1} ( \overline{B}\; ) $,  
 +
then its value at the centre of the ball is equal to the value of its arithmetical mean on the sphere $  S ( x _ {0} , R) $,  
 +
i.e.
  
<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/h/h046/h046470/h04647079.png" /></td> </tr></table>
+
$$
 +
u ( x _ {0} )  = \
 +
{
 +
\frac{1}{\sigma _ {n} ( R) }
 +
}
 +
\int\limits _ {S ( x _ {0} , R) }
 +
u ( y)  d \sigma ( y),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647080.png" /> is the surface area of the sphere of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647081.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647083.png" /> is only continuous, this property may be taken as the definition of a harmonic function.
+
where $  \sigma _ {n} ( R) $
 +
is the surface area of the sphere of radius $  R $
 +
in $  \mathbf R  ^ {n} $.  
 +
If $  u $
 +
is only continuous, this property may be taken as the definition of a harmonic function.
  
3) The maximum/minimum principle: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647084.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647085.png" /> not containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647086.png" /> as an interior point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647087.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647088.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647089.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647090.png" /> cannot attain a local extremum at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647091.png" />, i.e. in any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647092.png" /> of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647093.png" /> there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647094.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647095.png" /> and there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647096.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647097.png" /> (the maximum/minimum principle in local form). If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647098.png" />, then the largest and the least values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h04647099.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470100.png" /> are attained only at the points of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470101.png" /> (the maximum/minimum principle in global form). Consequently, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470103.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470104.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470105.png" />.
+
3) The maximum/minimum principle: Let $  D $
 +
be a domain in $  \mathbf R  ^ {n} $
 +
not containing $  \infty $
 +
as an interior point. If $  u $
 +
is a harmonic function in $  D $,  
 +
and $  u( x) \neq \textrm{ const } $,  
 +
then $  u $
 +
cannot attain a local extremum at any point $  x _ {0} \in D $,  
 +
i.e. in any neighbourhood $  V ( x _ {0} ) $
 +
of any point $  x _ {0} \in D $
 +
there exists a point $  x  ^ {*} \in V( x _ {0} ) $
 +
at which $  u ( x  ^ {*} ) > u ( x _ {0} ) $
 +
and there exists a point $  x  ^ {*} \in V ( x _ {0} ) $
 +
at which $  u ( x  ^ {*} ) < u ( x _ {0} ) $(
 +
the maximum/minimum principle in local form). If, in addition, $  u \in C ( \overline{D}\; ) $,  
 +
then the largest and the least values of $  u $
 +
on $  \overline{D}\; $
 +
are attained only at the points of the boundary $  \partial  D $(
 +
the maximum/minimum principle in global form). Consequently, if $  | u( x) | \leq  M $
 +
on $  \partial  D $,  
 +
then $  | u( x) | \leq  M $
 +
everywhere in $  \overline{D}\; $.
  
 
This principle may be generalized in various ways.
 
This principle may be generalized in various ways.
  
For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470106.png" /> is a harmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470107.png" /> not containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470108.png" /> and if
+
For instance, if $  u $
 +
is a harmonic function in a domain $  D $
 +
not containing $  \infty $
 +
and if
  
<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/h/h046/h046470/h046470109.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow y } \
 +
\sup  u ( x)  \leq  M
 +
$$
  
for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470110.png" /> (boundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470111.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470112.png" /> everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470113.png" />.
+
for all points $  y \in \partial  D $(
 +
boundary in $  \overline{ {\mathbf R  ^ {n} }}\; $),  
 +
then $  u( x) \leq  M $
 +
everywhere in $  D $.
  
4) The theorem on removable singularities: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470114.png" /> is a harmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470116.png" />, which satisfies the condition
+
4) The theorem on removable singularities: If $  u $
 +
is a harmonic function in a domain $  D \setminus  \{ x _ {0} \} $,  
 +
$  x _ {0} \in D $,  
 +
which satisfies the condition
  
<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/h/h046/h046470/h046470117.png" /></td> </tr></table>
+
$$
 +
u ( x)  = o ( | h _ {n} ( x - x _ {0} ) | ),\ \
 +
x \rightarrow x _ {0} ,
 +
$$
  
 
then there exists a finite limit
 
then there exists a finite limit
  
<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/h/h046/h046470/h046470118.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow x _ {0} } \
 +
u ( x)  = u ( x _ {0} ) ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470119.png" /> completed by the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470120.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470121.png" />.
+
and $  u $
 +
completed by the value $  u( x _ {0} ) $
 +
is a harmonic function in $  D $.
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470122.png" /> is a harmonic function throughout the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470124.png" />, bounded from above or from below, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470125.png" />.
+
5) If $  u $
 +
is a harmonic function throughout the space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
bounded from above or from below, then $  u = \textrm{ const } $.
  
6) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470126.png" /> is a harmonic function in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470127.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470128.png" /> can be expanded in this neighbourhood into a power series in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470129.png" />, i.e. all harmonic functions are analytic functions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470130.png" />; consequently, a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470131.png" /> has derivatives of all orders:
+
6) If $  u $
 +
is a harmonic function in a neighbourhood of a point $  x _ {0} = (( x _ {1} ) _ {0} \dots ( x _ {n} ) _ {0} ) $,  
 +
then $  u $
 +
can be expanded in this neighbourhood into a power series in the variables $  x _ {1} - ( x _ {1} ) _ {0} \dots x _ {n} - ( x _ {n} ) _ {0} $,  
 +
i.e. all harmonic functions are analytic functions of the variables $  x _ {1} \dots x _ {n} $;  
 +
consequently, a harmonic function $  u $
 +
has derivatives of all orders:
  
<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/h/h046/h046470/h046470132.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  ^ {m} u }{\partial  x _ {1} ^ {k _ {1} } \dots
 +
\partial  x _ {n} ^ {k _ {n} } }
 +
,\ \
 +
k _ {1} + \dots + k _ {n} = m,
 +
$$
  
 
which are also harmonic functions.
 
which are also harmonic functions.
  
7) The uniqueness property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470133.png" /> is a harmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470135.png" /> in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470136.png" />-dimensional neighbourhood of some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470137.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470138.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470139.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470140.png" /> is an analytic function of the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470141.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470142.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470143.png" /> is a harmonic function in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470144.png" />-dimensional neighbourhood of an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470145.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470146.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470147.png" />.
+
7) The uniqueness property: If $  u $
 +
is a harmonic function in a domain $  D \subset  \mathbf R  ^ {n} $
 +
and $  u \equiv 0 $
 +
in some $  n $-
 +
dimensional neighbourhood of some point $  x _ {0} \in D $,  
 +
then $  u \equiv 0 $
 +
in $  D $.  
 +
If $  u $
 +
is an analytic function of the real variables $  x = ( x _ {1} \dots x _ {n} ) $
 +
in a domain $  D \subset  \mathbf R  ^ {n} $
 +
and if $  u $
 +
is a harmonic function in some $  n $-
 +
dimensional neighbourhood of an arbitrary point $  x _ {0} \in D $,  
 +
then $  u $
 +
is a harmonic function in $  D $.
  
8) The symmetry principle: Let the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470148.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470149.png" /> contain a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470150.png" /> that is open in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470151.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470152.png" /> be a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470153.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470154.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470155.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470156.png" /> be the domain symmetrical to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470157.png" /> with respect to the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470158.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470159.png" /> can be harmonically extended into the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470160.png" /> by the formula
+
8) The symmetry principle: Let the boundary $  \partial  D $
 +
of a domain $  D \subset  \mathbf R  ^ {n} $
 +
contain a set $  G $
 +
that is open in the plane $  x _ {n} = 0 $,  
 +
let $  u $
 +
be a harmonic function in $  D $
 +
such that $  u = 0 $
 +
and continuous on $  G $
 +
and let $  \widetilde{D}  $
 +
be the domain symmetrical to $  D $
 +
with respect to the plane $  x _ {n} = 0 $;  
 +
then $  u $
 +
can be harmonically extended into the domain $  D \cup G \cup \widetilde{D}  $
 +
by the formula
  
<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/h/h046/h046470/h046470161.png" /></td> </tr></table>
+
$$
 +
u ( x _ {1} \dots x _ {n - 1 }  , x _ {n} )  = \
 +
- u ( x _ {1} \dots x _ {n - 1 }  , - x _ {n} ),
 +
$$
  
<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/h/h046/h046470/h046470162.png" /></td> </tr></table>
+
$$
 +
( x _ {1} \dots x _ {n - 1 }  , x _ {n} )  \in  \widetilde{D}  .
 +
$$
  
9) Harnack's first theorem: If a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470163.png" /> of harmonic functions in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470164.png" />, continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470165.png" />, converges uniformly on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470166.png" />, then it converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470167.png" />, and the limit function
+
9) Harnack's first theorem: If a sequence $  \{ u _ {n} \} $
 +
of harmonic functions in a bounded domain $  D $,  
 +
continuous in the closed domain $  \overline{D}\; $,  
 +
converges uniformly on the boundary $  \partial  D $,  
 +
then it converges uniformly on $  \overline{D}\; $,  
 +
and the limit function
  
<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/h/h046/h046470/h046470168.png" /></td> </tr></table>
+
$$
 +
u ( x)  = \
 +
\lim\limits _ {n \rightarrow \infty } \
 +
u _ {n} ( x)
 +
$$
  
is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470169.png" />.
+
is a harmonic function in $  D $.
  
10) Harnack's second theorem: If a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470170.png" /> of harmonic functions is monotone in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470171.png" /> and converges at least at one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470172.png" />, then it converges everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470173.png" /> towards a harmonic function
+
10) Harnack's second theorem: If a sequence $  \{ u _ {n} \} $
 +
of harmonic functions is monotone in a domain $  D $
 +
and converges at least at one point $  x _ {0} \in D $,  
 +
then it converges everywhere in $  D $
 +
towards a harmonic function
  
<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/h/h046/h046470/h046470174.png" /></td> </tr></table>
+
$$
 +
u ( x)  = \
 +
\lim\limits _ {n \rightarrow \infty } \
 +
u _ {n} ( x).
 +
$$
  
 
See also [[Harnack inequality|Harnack inequality]]; [[Harnack theorem|Harnack theorem]].
 
See also [[Harnack inequality|Harnack inequality]]; [[Harnack theorem|Harnack theorem]].
  
There exists a close connection between harmonic functions of two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470175.png" /> and analytic functions of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470176.png" />. The real and the imaginary part of an analytic function are, possibly multi-valued, conjugate harmonic functions, i.e. they are connected by the [[Cauchy-Riemann equations]]. If a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470177.png" /> is defined in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470178.png" />, the simplest way of finding an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470179.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470180.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470181.png" /> is given by the Goursat formula:
+
There exists a close connection between harmonic functions of two variables $  ( x _ {1} , x _ {2} ) $
 +
and analytic functions of the complex variable $  z = x _ {1} + i x _ {2} $.  
 +
The real and the imaginary part of an analytic function are, possibly multi-valued, conjugate harmonic functions, i.e. they are connected by the [[Cauchy-Riemann equations]]. If a harmonic function $  u ( x _ {1} , x _ {2} ) $
 +
is defined in a neighbourhood of a point $  ( x _ {1}  ^ {0} , x _ {2}  ^ {0} ) $,  
 +
the simplest way of finding an analytic function $  f( z) $,  
 +
$  z = x _ {1} + i x _ {2} $,  
 +
for which $  u ( x _ {1} , x _ {2} ) = \mathop{\rm Re}  f( z) $
 +
is given by the Goursat formula:
  
<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/h/h046/h046470/h046470182.png" /></td> </tr></table>
+
$$
 +
f ( z)  = 2u \left (
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470184.png" /> is an arbitrary real constant. Certain spatial problems in mathematical physics also involve multi-valued harmonic functions in domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470185.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470186.png" />.
+
\frac{z + \overline{ {z  ^ {0} }}\; }{2 }
 +
,\
  
The major importance of harmonic functions in mathematical physics is mainly due to the frequent occurrence of vector fields of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470187.png" />. Such fields in domains not containing field sources must satisfy the conservation equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470188.png" />, i.e. the Laplace equation, which means that in such domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470189.png" /> is a harmonic function.
+
\frac{z - \overline{ {z  ^ {0} }}\; }{2i }
 +
\right )
 +
- u ( x _ {1}  ^ {0} , x _ {2}  ^ {0} ) + iC _ {0} ,
 +
$$
  
Examples. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470190.png" /> is the force vector of the gravity field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470191.png" /> is the Newton potential of the gravitational forces; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470192.png" /> is the field of velocities of a stationary motion of an incompressible homogeneous gas or liquid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470193.png" /> is the velocity potential; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470194.png" /> is the strength of an electrostatic field in a homogeneous isotropic medium, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470195.png" /> is the potential of the electrostatic field; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470196.png" /> is the strength of a stationary magnetic field in a homogeneous isotropic medium, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470197.png" /> is the scalar, usually multi-valued, potential of the magnetic field. In the case of steady propagation of heat in a uniform isotropic medium or a stationary distribution of diffusing particles, the harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470198.png" /> is the temperature of the medium or the density of the particles at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470199.png" />, respectively. Many important problems in the theory of elasticity and in the theory of electromagnetic fields can also be reduced to solving problems concerning harmonic functions.
+
where  $  \overline{ {z  ^ {0} }}\; = x _ {1}  ^ {0} - i x _ {2}  ^ {0} $
 +
and  $  C _ {0} $
 +
is an arbitrary real constant. Certain spatial problems in mathematical physics also involve multi-valued harmonic functions in domains in $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $.
  
The boundary [[Dirichlet problem|Dirichlet problem]], or the first boundary value problem, is of special importance in the development of the theory of harmonic functions and mathematical physics. It consists in finding a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470200.png" /> which is harmonic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470201.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470202.png" />, from given continuous values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470203.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470204.png" />. If the surface or line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470205.png" /> is sufficiently smooth, the solution may be expressed by the [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470206.png" />:
+
The major importance of harmonic functions in mathematical physics is mainly due to the frequent occurrence of vector fields of the form  $  \mathbf s = -  \mathop{\rm grad}  u $.  
 +
Such fields in domains not containing field sources must satisfy the conservation equation  $  \mathop{\rm div}  \mathbf s = - \Delta u = 0 $,  
 +
i.e. the Laplace equation, which means that in such domains  $  u $
 +
is a harmonic function.
  
<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/h/h046/h046470/h046470207.png" /></td> </tr></table>
+
Examples. If  $  \mathbf s $
 +
is the force vector of the gravity field,  $  u $
 +
is the Newton potential of the gravitational forces; if  $  \mathbf s $
 +
is the field of velocities of a stationary motion of an incompressible homogeneous gas or liquid,  $  u $
 +
is the velocity potential; if  $  \mathbf s $
 +
is the strength of an electrostatic field in a homogeneous isotropic medium,  $  u $
 +
is the potential of the electrostatic field; if  $  \mathbf s $
 +
is the strength of a stationary magnetic field in a homogeneous isotropic medium,  $  u $
 +
is the scalar, usually multi-valued, potential of the magnetic field. In the case of steady propagation of heat in a uniform isotropic medium or a stationary distribution of diffusing particles, the harmonic function  $  u( x) $
 +
is the temperature of the medium or the density of the particles at a point  $  x $,
 +
respectively. Many important problems in the theory of elasticity and in the theory of electromagnetic fields can also be reduced to solving problems concerning harmonic functions.
  
In the case of the simplest domains (spheres, half-spaces), when the normal derivative is readily expressed in explicit form, the [[Poisson integral|Poisson integral]] is obtained. The second boundary value problem, or the [[Neumann problem]], is also often encountered. It consists in determining a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470208.png" /> from given values of its normal derivative on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470209.png" />. This problem can be solved using the corresponding Green function, but explicit expressions are much more complicated in this case. There are many more boundary value problems in the theory of harmonic functions, the formulations and solutions of which are more complicated. See also [[Balayage method|Balayage method]]; [[Robin problem|Robin problem]].
+
The boundary [[Dirichlet problem|Dirichlet problem]], or the first boundary value problem, is of special importance in the development of the theory of harmonic functions and mathematical physics. It consists in finding a function  $  u $
 +
which is harmonic in a domain  $  D $
 +
and continuous on  $  \overline{D}\; $,
 +
from given continuous values $  u( y) $
 +
on the boundary $  S = \partial  D $.  
 +
If the surface or line  $  S $
 +
is sufficiently smooth, the solution may be expressed by the [[Green function|Green function]] $  G( x, y) $:
  
A special place in the modern theory of harmonic functions is occupied by ill-posed problems, mainly those connected with the Cauchy problem for the Laplace equation. These include, for example, the following problem on the best majorant: If on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470210.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470211.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470212.png" /> and the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470213.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470214.png" /> are given, find the best possible estimate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470215.png" /> in the class of harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470216.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470217.png" /> [[#References|[9]]], [[#References|[10]]].
+
$$
 +
u ( x)  = \
 +
- \int\limits _ { S }
 +
u ( y)
  
The study of the boundary properties of harmonic functions related with subharmonic functions (cf. [[Subharmonic function|Subharmonic function]]) and with the [[Boundary properties of analytic functions|boundary properties of analytic functions]] is of importance. For instance, a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470218.png" /> in the unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470219.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470220.png" /> usually has no radial limit values
+
\frac{\partial  G ( x, y) }{\partial  \nu _ {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/h/h046/h046470/h046470221.png" /></td> </tr></table>
+
In the case of the simplest domains (spheres, half-spaces), when the normal derivative is readily expressed in explicit form, the [[Poisson integral|Poisson integral]] is obtained. The second boundary value problem, or the [[Neumann problem]], is also often encountered. It consists in determining a harmonic function  $  u $
 +
from given values of its normal derivative on the boundary  $  S $.
 +
This problem can be solved using the corresponding Green function, but explicit expressions are much more complicated in this case. There are many more boundary value problems in the theory of harmonic functions, the formulations and solutions of which are more complicated. See also [[Balayage method|Balayage method]]; [[Robin problem|Robin problem]].
  
However, in the case of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470222.png" /> of harmonic functions defined by the condition
+
A special place in the modern theory of harmonic functions is occupied by ill-posed problems, mainly those connected with the Cauchy problem for the Laplace equation. These include, for example, the following problem on the best majorant: If on the boundary  $  S = \partial  D $
 +
of a domain  $  D $
 +
the function  $  M = M( y) $
 +
and the conditions  $  | u( y) | \leq  M ( y) $,
 +
$  | \partial  u ( y) / \partial  \nu | \leq  M ( y) $
 +
are given, find the best possible estimate of  $  \sup  | u( x) | $
 +
in the class of harmonic functions $  u $
 +
in  $  D $[[#References|[9]]], [[#References|[10]]].
  
<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/h/h046/h046470/h046470223.png" /></td> </tr></table>
+
The study of the boundary properties of harmonic functions related with subharmonic functions (cf. [[Subharmonic function|Subharmonic function]]) and with the [[Boundary properties of analytic functions|boundary properties of analytic functions]] is of importance. For instance, a harmonic function  $  u $
 +
in the unit ball  $  B ( 0, 1 ) $
 +
of  $  \mathbf R  ^ {n} $
 +
usually has no radial limit values
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470224.png" /> is the surface element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470225.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470226.png" />, the radial boundary values exist almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470227.png" /> with respect to Lebesgue measure, and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470228.png" /> can be represented in the form of a Poisson–Stieltjes integral
+
$$
 +
f ( y)  = \
 +
\lim\limits _ {r \uparrow 1 }  u ( ry),\ \
 +
y \in S = \partial  B ( 0, 1).
 +
$$
  
<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/h/h046/h046470/h046470229.png" /></td> </tr></table>
+
However, in the case of the class $  A $
 +
of harmonic functions defined by the condition
 +
 
 +
$$
 +
\int\limits _ { S } u  ^ {+}
 +
( ry)  d \sigma ( y)  \leq  \
 +
C ( u)  < \infty ,
 +
$$
 +
 
 +
where  $  d \sigma ( y) $
 +
is the surface element of  $  S $,
 +
$  u  ^ {+} = \max \{ 0, u \} $,
 +
the radial boundary values exist almost-everywhere on  $  S $
 +
with respect to Lebesgue measure, and an  $  u \in A $
 +
can be represented in the form of a Poisson–Stieltjes integral
 +
 
 +
$$
 +
u ( x)  = \int\limits _ { S }
 +
P _ {n} ( x, y) \
 +
d \mu ( y),
 +
$$
  
 
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/h/h046/h046470/h046470230.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( x, y)  = \
 +
{
 +
\frac{1}{\sigma _ {n} ( 1) }
 +
}
  
is the Poisson kernel and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470231.png" /> is the Borel measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470232.png" />. The proper subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470233.png" /> of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470234.png" /> consisting of all harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470235.png" /> that can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046470/h046470236.png" /> by a Poisson–Lebesgue integral,
+
\frac{1 - | x |  ^ {2} }{| x - y |  ^ {n} }
  
<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/h/h046/h046470/h046470237.png" /></td> </tr></table>
+
$$
 +
 
 +
is the Poisson kernel and  $  d \mu $
 +
is the Borel measure on  $  S $.
 +
The proper subclass  $  B $
 +
of the class $  A $
 +
consisting of all harmonic functions  $  u $
 +
that can be represented in  $  B ( 0, 1) $
 +
by a Poisson–Lebesgue integral,
 +
 
 +
$$
 +
u ( x)  = \int\limits _ { S }
 +
P _ {n} ( x, y) f ( y)  d \sigma ( y),
 +
$$
  
 
is also of importance.
 
is also of importance.
Line 141: Line 493:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.F. Timan, V.N. Trofimov, "Introduction to the theory of harmonic functions" , Moscow (1968) (In Russian) {{MR|0315142}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.M. [N.M. Gyunter] Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian) {{MR|0222316}} {{ZBL|0164.41901}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.N. Sretenskii, "Theory of the Newton potential" , Moscow-Leningrad (1946) (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) {{MR|0106366}} {{ZBL|0084.30903}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> O.D. Kellogg, "Foundations of potential theory" , Springer (1967) (Re-issue: Springer, 1967) {{MR|0222317}} {{ZBL|0152.31301}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) {{MR|0764399}} {{ZBL|0954.35001}} {{ZBL|0652.35002}} {{ZBL|0695.35001}} {{ZBL|0699.35005}} {{ZBL|0607.35001}} {{ZBL|0506.35001}} {{ZBL|0223.35002}} {{ZBL|0231.35002}} {{ZBL|0207.09101}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> M.M. [M.M. Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics" , Springer (1967) (Translated from Russian) {{MR|}} {{ZBL|0149.41902}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> S.N. Mergelyan, "Harmonic approximation and approximate solution of Cauchy's problem for the Laplace equation" ''Uspekhi Mat. Nauk'' , '''11''' : 5 (1956) pp. 3–26 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) {{MR|0083565}} {{ZBL|}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" ''Itogi Nauk. Ser. Mat., Mat. Anal., Teor. Veroyatnost., Regulirovanie, 1962'' (1964) pp. 83–100 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.F. Timan, V.N. Trofimov, "Introduction to the theory of harmonic functions" , Moscow (1968) (In Russian) {{MR|0315142}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.M. [N.M. Gyunter] Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian) {{MR|0222316}} {{ZBL|0164.41901}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.N. Sretenskii, "Theory of the Newton potential" , Moscow-Leningrad (1946) (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) {{MR|0106366}} {{ZBL|0084.30903}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> O.D. Kellogg, "Foundations of potential theory" , Springer (1967) (Re-issue: Springer, 1967) {{MR|0222317}} {{ZBL|0152.31301}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) {{MR|0764399}} {{ZBL|0954.35001}} {{ZBL|0652.35002}} {{ZBL|0695.35001}} {{ZBL|0699.35005}} {{ZBL|0607.35001}} {{ZBL|0506.35001}} {{ZBL|0223.35002}} {{ZBL|0231.35002}} {{ZBL|0207.09101}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> M.M. [M.M. Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics" , Springer (1967) (Translated from Russian) {{MR|}} {{ZBL|0149.41902}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> S.N. Mergelyan, "Harmonic approximation and approximate solution of Cauchy's problem for the Laplace equation" ''Uspekhi Mat. Nauk'' , '''11''' : 5 (1956) pp. 3–26 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) {{MR|0083565}} {{ZBL|}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" ''Itogi Nauk. Ser. Mat., Mat. Anal., Teor. Veroyatnost., Regulirovanie, 1962'' (1964) pp. 83–100 (In Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:43, 5 June 2020


A real-valued function $ u $, defined in a domain $ D $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, having continuous partial derivatives of the first and second orders in $ D $, and which is a solution of the Laplace equation

$$ \Delta u \equiv \ \frac{\partial ^ {2} u }{\partial x _ {1} ^ {2} } + \dots + \frac{\partial ^ {2} u }{\partial x _ {n} ^ {2} } = 0, $$

where $ x _ {1} \dots x _ {n} $ are the orthogonal Cartesian coordinates of the point $ x $. This definition is sometimes extended to include complex functions $ w( x) = u( x) + iv( x) $ as well, in the sense that their real and imaginary parts $ \mathop{\rm Re} w( x) = u ( x) $ and $ \mathop{\rm Im} w ( x) = v ( x) $ are harmonic functions. The requirements of continuity and even of the existence of derivatives are not a priori indispensable. For instance, one of Privalov's theorems is applicable: A continuous function $ u $ in $ D $ is a harmonic function if and only if at any point $ x \in D $ the mean-value property

$$ u ( x) = \ { \frac{1}{\omega _ {n} ( R) } } \int\limits _ {B _ {n} ( x, R) } u ( y) dy $$

— where $ B _ {n} ( x, R) $ is the ball of radius $ R $ with centre at $ x $, $ \omega _ {n} ( R) $ is the volume of this ball and $ dy $ is the volume element in $ \mathbf R ^ {n} $— is fulfilled for sufficiently small $ R > 0 $.

If $ D $ is unbounded with a compact boundary $ \partial D $, the definition of a harmonic function may be completed to include the point at infinity $ \infty $, i.e. it may additionally be defined in domains in the Aleksandrov compactification of $ \mathbf R ^ {n} $. The general principle of such a completion of the definition is that, under the simplest transformations which preserve harmonicity (inversion if $ n = 2 $, Kelvin transformations if $ n \geq 3 $, cf. Kelvin transformation) and map a finite point $ x _ {0} $ into $ \infty $, a harmonic function in a neighbourhood of $ x _ {0} $ becomes a harmonic function in a neighbourhood of $ \infty $. On this basis, a harmonic function $ u $ is said to be regular at infinity for $ n \geq 3 $ if

$$ \lim\limits _ {| x | \rightarrow \infty } \ u ( x) = 0,\ \ | x | = \sqrt {x _ {1} ^ {2} + \dots + x _ {n} ^ {2} } . $$

Thus, for a harmonic function $ u $ which is regular at infinity one always has $ u( \infty ) = 0 $ if $ n \geq 3 $. If $ n = 2 $, the condition

$$ u ( x) = O( 1),\ | x | \rightarrow \infty , $$

which implies the existence of a finite limit

$$ \lim\limits _ {| x | \rightarrow \infty } \ u ( x) = u ( \infty ), $$

must be met. Harmonic functions in unbounded domains are usually understood to mean harmonic functions regular at infinity.

In the theory of harmonic functions an important role is played by the principal fundamental solutions of the Laplace equation:

$$ h _ {2} ( x) = \ { \frac{1}{2 \pi } } \ \mathop{\rm ln} { \frac{1}{| x | } } \ \ \textrm{ if } \ n = 2, $$

$$ h _ {n} ( x) = { \frac{1}{( n - 2) \sigma _ {n} } } { \frac{1}{| x | ^ {n - 2 } } } \ \textrm{ if } n \geq 3, $$

where $ \sigma _ {n} $ is the surface area of the unit sphere in $ \mathbf R ^ {n} $. If $ | x | > 0 $, this is a harmonic function. The fundamental solutions may be used to write down the basic formula of the theory of harmonic functions, which expresses the values of a harmonic function $ u ( x) $ inside a domain $ D $ in terms of its values $ u( y) $ on the boundary $ S = \partial D $ and the values of its derivative in the direction of the exterior normal $ \partial u ( y) / \partial \nu $ towards $ S $ at the point $ y $:

$$ \int\limits _ { S } \left [ h _ {n} ( x - y) \frac{\partial u ( y) }{\partial \nu } - u ( y) \frac{\partial h _ {n} ( x - y) }{\partial \nu _ {y} } \right ] d \sigma ( y) = $$

$$ = \ \left \{ \begin{array}{ll} u ( x), & x \in D, \\ 0, & x \notin \overline{D}\; . \\ \end{array} \right .$$

This Green formula is valid, for example, if the function $ u $ and its partial derivatives of the first order are continuous in the closed domain $ \overline{D}\; $, i.e. if $ u \in C ^ {1} ( \overline{D}\; ) $, the boundary $ S $ of which is a piecewise-smooth closed surface or curve. It yields a representation of an arbitrary harmonic function $ u $ as the sum of single- and double-layer potentials (cf. Potential theory). The densities of these potentials, i.e. the boundary values $ \partial u ( y) / \partial \nu $ and $ u( y) $ respectively, cannot be specified arbitrarily. There is an integral relationship between the two, in that the left-hand side of the last-named formula — the Green integral — must vanish for all points $ x $ outside $ \overline{D}\; $. The basic formula of the theory of harmonic functions is a direct analogue of the fundamental formula of the theory of analytic functions — the integral formula of Cauchy (cf. Cauchy integral). This formula also remains valid if the principal fundamental solution $ h _ {n} $ in it is replaced by any other fundamental solution of the Laplace equation which is sufficiently smooth in $ \overline{D}\; $, e.g. belongs to $ C ^ {1} ( \overline{D}\; ) $.

The fundamental properties of harmonic functions, on the assumption that the boundary $ S $ of the domain $ D $ is piecewise smooth, are listed below. After suitable modification, many of them are also valid for complex harmonic functions.

1) If $ D $ is a bounded domain and a harmonic function $ u \in C ^ {1} ( \overline{D}\; ) $, then

$$ \int\limits _ { S } \frac{\partial u ( y) }{\partial \nu } \ d \sigma ( y) = 0. $$

2) The mean-value theorem: If $ u $ is a harmonic function in the ball $ B = B ( x _ {0,\ } R) $ of radius $ R $ with centre at $ x _ {0} $ and if $ u \in C ^ {1} ( \overline{B}\; ) $, then its value at the centre of the ball is equal to the value of its arithmetical mean on the sphere $ S ( x _ {0} , R) $, i.e.

$$ u ( x _ {0} ) = \ { \frac{1}{\sigma _ {n} ( R) } } \int\limits _ {S ( x _ {0} , R) } u ( y) d \sigma ( y), $$

where $ \sigma _ {n} ( R) $ is the surface area of the sphere of radius $ R $ in $ \mathbf R ^ {n} $. If $ u $ is only continuous, this property may be taken as the definition of a harmonic function.

3) The maximum/minimum principle: Let $ D $ be a domain in $ \mathbf R ^ {n} $ not containing $ \infty $ as an interior point. If $ u $ is a harmonic function in $ D $, and $ u( x) \neq \textrm{ const } $, then $ u $ cannot attain a local extremum at any point $ x _ {0} \in D $, i.e. in any neighbourhood $ V ( x _ {0} ) $ of any point $ x _ {0} \in D $ there exists a point $ x ^ {*} \in V( x _ {0} ) $ at which $ u ( x ^ {*} ) > u ( x _ {0} ) $ and there exists a point $ x ^ {*} \in V ( x _ {0} ) $ at which $ u ( x ^ {*} ) < u ( x _ {0} ) $( the maximum/minimum principle in local form). If, in addition, $ u \in C ( \overline{D}\; ) $, then the largest and the least values of $ u $ on $ \overline{D}\; $ are attained only at the points of the boundary $ \partial D $( the maximum/minimum principle in global form). Consequently, if $ | u( x) | \leq M $ on $ \partial D $, then $ | u( x) | \leq M $ everywhere in $ \overline{D}\; $.

This principle may be generalized in various ways.

For instance, if $ u $ is a harmonic function in a domain $ D $ not containing $ \infty $ and if

$$ \lim\limits _ {x \rightarrow y } \ \sup u ( x) \leq M $$

for all points $ y \in \partial D $( boundary in $ \overline{ {\mathbf R ^ {n} }}\; $), then $ u( x) \leq M $ everywhere in $ D $.

4) The theorem on removable singularities: If $ u $ is a harmonic function in a domain $ D \setminus \{ x _ {0} \} $, $ x _ {0} \in D $, which satisfies the condition

$$ u ( x) = o ( | h _ {n} ( x - x _ {0} ) | ),\ \ x \rightarrow x _ {0} , $$

then there exists a finite limit

$$ \lim\limits _ {x \rightarrow x _ {0} } \ u ( x) = u ( x _ {0} ) , $$

and $ u $ completed by the value $ u( x _ {0} ) $ is a harmonic function in $ D $.

5) If $ u $ is a harmonic function throughout the space $ \mathbf R ^ {n} $, $ n \geq 2 $, bounded from above or from below, then $ u = \textrm{ const } $.

6) If $ u $ is a harmonic function in a neighbourhood of a point $ x _ {0} = (( x _ {1} ) _ {0} \dots ( x _ {n} ) _ {0} ) $, then $ u $ can be expanded in this neighbourhood into a power series in the variables $ x _ {1} - ( x _ {1} ) _ {0} \dots x _ {n} - ( x _ {n} ) _ {0} $, i.e. all harmonic functions are analytic functions of the variables $ x _ {1} \dots x _ {n} $; consequently, a harmonic function $ u $ has derivatives of all orders:

$$ \frac{\partial ^ {m} u }{\partial x _ {1} ^ {k _ {1} } \dots \partial x _ {n} ^ {k _ {n} } } ,\ \ k _ {1} + \dots + k _ {n} = m, $$

which are also harmonic functions.

7) The uniqueness property: If $ u $ is a harmonic function in a domain $ D \subset \mathbf R ^ {n} $ and $ u \equiv 0 $ in some $ n $- dimensional neighbourhood of some point $ x _ {0} \in D $, then $ u \equiv 0 $ in $ D $. If $ u $ is an analytic function of the real variables $ x = ( x _ {1} \dots x _ {n} ) $ in a domain $ D \subset \mathbf R ^ {n} $ and if $ u $ is a harmonic function in some $ n $- dimensional neighbourhood of an arbitrary point $ x _ {0} \in D $, then $ u $ is a harmonic function in $ D $.

8) The symmetry principle: Let the boundary $ \partial D $ of a domain $ D \subset \mathbf R ^ {n} $ contain a set $ G $ that is open in the plane $ x _ {n} = 0 $, let $ u $ be a harmonic function in $ D $ such that $ u = 0 $ and continuous on $ G $ and let $ \widetilde{D} $ be the domain symmetrical to $ D $ with respect to the plane $ x _ {n} = 0 $; then $ u $ can be harmonically extended into the domain $ D \cup G \cup \widetilde{D} $ by the formula

$$ u ( x _ {1} \dots x _ {n - 1 } , x _ {n} ) = \ - u ( x _ {1} \dots x _ {n - 1 } , - x _ {n} ), $$

$$ ( x _ {1} \dots x _ {n - 1 } , x _ {n} ) \in \widetilde{D} . $$

9) Harnack's first theorem: If a sequence $ \{ u _ {n} \} $ of harmonic functions in a bounded domain $ D $, continuous in the closed domain $ \overline{D}\; $, converges uniformly on the boundary $ \partial D $, then it converges uniformly on $ \overline{D}\; $, and the limit function

$$ u ( x) = \ \lim\limits _ {n \rightarrow \infty } \ u _ {n} ( x) $$

is a harmonic function in $ D $.

10) Harnack's second theorem: If a sequence $ \{ u _ {n} \} $ of harmonic functions is monotone in a domain $ D $ and converges at least at one point $ x _ {0} \in D $, then it converges everywhere in $ D $ towards a harmonic function

$$ u ( x) = \ \lim\limits _ {n \rightarrow \infty } \ u _ {n} ( x). $$

See also Harnack inequality; Harnack theorem.

There exists a close connection between harmonic functions of two variables $ ( x _ {1} , x _ {2} ) $ and analytic functions of the complex variable $ z = x _ {1} + i x _ {2} $. The real and the imaginary part of an analytic function are, possibly multi-valued, conjugate harmonic functions, i.e. they are connected by the Cauchy-Riemann equations. If a harmonic function $ u ( x _ {1} , x _ {2} ) $ is defined in a neighbourhood of a point $ ( x _ {1} ^ {0} , x _ {2} ^ {0} ) $, the simplest way of finding an analytic function $ f( z) $, $ z = x _ {1} + i x _ {2} $, for which $ u ( x _ {1} , x _ {2} ) = \mathop{\rm Re} f( z) $ is given by the Goursat formula:

$$ f ( z) = 2u \left ( \frac{z + \overline{ {z ^ {0} }}\; }{2 } ,\ \frac{z - \overline{ {z ^ {0} }}\; }{2i } \right ) - u ( x _ {1} ^ {0} , x _ {2} ^ {0} ) + iC _ {0} , $$

where $ \overline{ {z ^ {0} }}\; = x _ {1} ^ {0} - i x _ {2} ^ {0} $ and $ C _ {0} $ is an arbitrary real constant. Certain spatial problems in mathematical physics also involve multi-valued harmonic functions in domains in $ \mathbf R ^ {n} $, $ n \geq 2 $.

The major importance of harmonic functions in mathematical physics is mainly due to the frequent occurrence of vector fields of the form $ \mathbf s = - \mathop{\rm grad} u $. Such fields in domains not containing field sources must satisfy the conservation equation $ \mathop{\rm div} \mathbf s = - \Delta u = 0 $, i.e. the Laplace equation, which means that in such domains $ u $ is a harmonic function.

Examples. If $ \mathbf s $ is the force vector of the gravity field, $ u $ is the Newton potential of the gravitational forces; if $ \mathbf s $ is the field of velocities of a stationary motion of an incompressible homogeneous gas or liquid, $ u $ is the velocity potential; if $ \mathbf s $ is the strength of an electrostatic field in a homogeneous isotropic medium, $ u $ is the potential of the electrostatic field; if $ \mathbf s $ is the strength of a stationary magnetic field in a homogeneous isotropic medium, $ u $ is the scalar, usually multi-valued, potential of the magnetic field. In the case of steady propagation of heat in a uniform isotropic medium or a stationary distribution of diffusing particles, the harmonic function $ u( x) $ is the temperature of the medium or the density of the particles at a point $ x $, respectively. Many important problems in the theory of elasticity and in the theory of electromagnetic fields can also be reduced to solving problems concerning harmonic functions.

The boundary Dirichlet problem, or the first boundary value problem, is of special importance in the development of the theory of harmonic functions and mathematical physics. It consists in finding a function $ u $ which is harmonic in a domain $ D $ and continuous on $ \overline{D}\; $, from given continuous values $ u( y) $ on the boundary $ S = \partial D $. If the surface or line $ S $ is sufficiently smooth, the solution may be expressed by the Green function $ G( x, y) $:

$$ u ( x) = \ - \int\limits _ { S } u ( y) \frac{\partial G ( x, y) }{\partial \nu _ {y} } \ d \sigma ( y). $$

In the case of the simplest domains (spheres, half-spaces), when the normal derivative is readily expressed in explicit form, the Poisson integral is obtained. The second boundary value problem, or the Neumann problem, is also often encountered. It consists in determining a harmonic function $ u $ from given values of its normal derivative on the boundary $ S $. This problem can be solved using the corresponding Green function, but explicit expressions are much more complicated in this case. There are many more boundary value problems in the theory of harmonic functions, the formulations and solutions of which are more complicated. See also Balayage method; Robin problem.

A special place in the modern theory of harmonic functions is occupied by ill-posed problems, mainly those connected with the Cauchy problem for the Laplace equation. These include, for example, the following problem on the best majorant: If on the boundary $ S = \partial D $ of a domain $ D $ the function $ M = M( y) $ and the conditions $ | u( y) | \leq M ( y) $, $ | \partial u ( y) / \partial \nu | \leq M ( y) $ are given, find the best possible estimate of $ \sup | u( x) | $ in the class of harmonic functions $ u $ in $ D $[9], [10].

The study of the boundary properties of harmonic functions related with subharmonic functions (cf. Subharmonic function) and with the boundary properties of analytic functions is of importance. For instance, a harmonic function $ u $ in the unit ball $ B ( 0, 1 ) $ of $ \mathbf R ^ {n} $ usually has no radial limit values

$$ f ( y) = \ \lim\limits _ {r \uparrow 1 } u ( ry),\ \ y \in S = \partial B ( 0, 1). $$

However, in the case of the class $ A $ of harmonic functions defined by the condition

$$ \int\limits _ { S } u ^ {+} ( ry) d \sigma ( y) \leq \ C ( u) < \infty , $$

where $ d \sigma ( y) $ is the surface element of $ S $, $ u ^ {+} = \max \{ 0, u \} $, the radial boundary values exist almost-everywhere on $ S $ with respect to Lebesgue measure, and an $ u \in A $ can be represented in the form of a Poisson–Stieltjes integral

$$ u ( x) = \int\limits _ { S } P _ {n} ( x, y) \ d \mu ( y), $$

where

$$ P _ {n} ( x, y) = \ { \frac{1}{\sigma _ {n} ( 1) } } \frac{1 - | x | ^ {2} }{| x - y | ^ {n} } $$

is the Poisson kernel and $ d \mu $ is the Borel measure on $ S $. The proper subclass $ B $ of the class $ A $ consisting of all harmonic functions $ u $ that can be represented in $ B ( 0, 1) $ by a Poisson–Lebesgue integral,

$$ u ( x) = \int\limits _ { S } P _ {n} ( x, y) f ( y) d \sigma ( y), $$

is also of importance.

Substantial advances have been made in the axiomatic theory of harmonic functions and potentials in topological spaces (cf. Harmonic space; Potential theory, abstract).

References

[1] A.F. Timan, V.N. Trofimov, "Introduction to the theory of harmonic functions" , Moscow (1968) (In Russian) MR0315142
[2] N.M. [N.M. Gyunter] Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian) MR0222316 Zbl 0164.41901
[3] L.N. Sretenskii, "Theory of the Newton potential" , Moscow-Leningrad (1946) (In Russian)
[4] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903
[5] O.D. Kellogg, "Foundations of potential theory" , Springer (1967) (Re-issue: Springer, 1967) MR0222317 Zbl 0152.31301
[6] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101
[7] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[8] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[9] M.M. [M.M. Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics" , Springer (1967) (Translated from Russian) Zbl 0149.41902
[10] S.N. Mergelyan, "Harmonic approximation and approximate solution of Cauchy's problem for the Laplace equation" Uspekhi Mat. Nauk , 11 : 5 (1956) pp. 3–26 (In Russian)
[11] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) MR0083565
[12] E.D. Solomentsev, "Harmonic and subharmonic functions and their generalizations" Itogi Nauk. Ser. Mat., Mat. Anal., Teor. Veroyatnost., Regulirovanie, 1962 (1964) pp. 83–100 (In Russian)

Comments

Fundamental property 5) above is also called Picard's theorem (for harmonic functions).

The symmetry principle is also known as the Schwarz reflection principle (cf. Schwarz symmetry theorem). [a1] defines harmonic functions with respect to the fine topology.

References

[a1] B. Fuglede, "Finely harmonic functions" , Springer (1972) MR0450590 Zbl 0248.31010
[a2] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976) MR0460672 MR0419791 MR0412442 MR0442324 Zbl 0419.31001 Zbl 0339.31003 Zbl 0328.33011
[a3] L.L. Helms, "Introduction to potential theory" , Wiley (Interscience) (1969)
How to Cite This Entry:
Harmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_function&oldid=47180
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article