Namespaces
Variants
Actions

Difference between revisions of "Green formulas"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
Line 1: Line 1:
Formulas of the integral calculus of functions of several variables, connecting the values of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450801.png" />-fold integral over a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450802.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450803.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450804.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450805.png" />-fold integral along the piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450806.png" /> of this domain. The Green formulas are obtained by integration by parts of integrals of the divergence of a vector field that is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450807.png" /> and that is continuously differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g0450808.png" />.
+
<!--
 +
g0450801.png
 +
$#A+1 = 101 n = 0
 +
$#C+1 = 101 : ~/encyclopedia/old_files/data/G045/G.0405080 Green formulas
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
Formulas of the integral calculus of functions of several variables, connecting the values of an $  n $-
 +
fold integral over a domain $  D $
 +
in an $  n $-
 +
dimensional Euclidean space $  E  ^ {n} $
 +
with an $  ( n - 1) $-
 +
fold integral along the piecewise-smooth boundary $  \partial  D = \Gamma $
 +
of this domain. The Green formulas are obtained by integration by parts of integrals of the divergence of a vector field that is continuous in $  \overline{D}\; = D + \Gamma $
 +
and that is continuously differentiable in $  D $.
  
 
In the simplest Green formula,
 
In the simplest Green 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/g/g045/g045080/g0450809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\int\limits _  \Gamma  [ P  dx + Q  dy]  = \
 +
\int\limits \int\limits _ { D }
 +
\left [
 +
 
 +
\frac{\partial  Q }{\partial  x }
 +
-
 +
 
 +
\frac{\partial  P }{\partial  y }
  
the curvilinear integral along the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508010.png" /> is expressed as a double integral over the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508011.png" />. Here the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508012.png" /> is oriented in a natural manner, while an induced orientation, known as counterclockwise, is taken along the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508013.png" />. Formula (1) has a simple hydrodynamic meaning: The flow across the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508014.png" /> of a liquid flowing on a plane at rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508015.png" /> is equal to the integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508016.png" /> of the intensity (divergence) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508017.png" /> of the sources and sinks distributed over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508018.png" />. In this sense the Green formula (1) resembles the [[Ostrogradski formula|Ostrogradski formula]] (see also [[Stokes formula|Stokes formula]]).
+
\right ]  dx  dy,
 +
$$
 +
 
 +
the curvilinear integral along the contour $  \Gamma $
 +
is expressed as a double integral over the domain $  D \subset  E  ^ {2} $.  
 +
Here the domain $  D $
 +
is oriented in a natural manner, while an induced orientation, known as counterclockwise, is taken along the boundary $  \Gamma $.  
 +
Formula (1) has a simple hydrodynamic meaning: The flow across the boundary $  \Gamma $
 +
of a liquid flowing on a plane at rate $  \mathbf v = ( Q, - P) $
 +
is equal to the integral over $  D $
 +
of the intensity (divergence) $  \mathop{\rm div}  \mathbf v = ( \partial  Q/ \partial  x) - ( \partial  P/ \partial  y) $
 +
of the sources and sinks distributed over $  D $.  
 +
In this sense the Green formula (1) resembles the [[Ostrogradski formula|Ostrogradski formula]] (see also [[Stokes formula|Stokes formula]]).
  
 
Formula (1) is sometimes attributed to C.F. Gauss and B. Riemann, but none of its usual appellations corresponds to historical truth; it was in fact encountered as early as the 18th century in the analytical studies of L. Euler and others.
 
Formula (1) is sometimes attributed to C.F. Gauss and B. Riemann, but none of its usual appellations corresponds to historical truth; it was in fact encountered as early as the 18th century in the analytical studies of L. Euler and others.
Line 11: Line 50:
 
G. Green [[#References|[1]]] must be credited with the following formulas of [[Potential theory|potential theory]]:
 
G. Green [[#References|[1]]] must be credited with the following formulas of [[Potential theory|potential theory]]:
  
<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/g/g045/g045080/g04508019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { D }
 +
\left [
 +
v \Delta u +
 +
\sum _ {i = 1 } ^ { 3 }
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {i} }
 +
 
 +
\frac{\partial  v }{\partial  x  ^ {i} }
 +
\right ]  dx  = \
 +
\int\limits _  \Gamma  v
 +
 
 +
\frac{\partial  u }{\partial  N }
 +
  ds,
 +
$$
  
 
the preparatory Green formula, and
 
the preparatory Green formula, and
  
<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/g/g045/g045080/g04508020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\int\limits _ { D }
 +
[ u \Delta v - v \Delta u]  dx  = \
 +
\int\limits _  \Gamma  \left [
 +
u
 +
\frac{\partial  v }{\partial  N }
 +
-
 +
v
 +
\frac{\partial  u }{\partial  N }
 +
 
 +
\right ]  ds.
 +
$$
 +
 
 +
Here  $  D $
 +
is a domain in  $  E  ^ {3} $,
 +
$  x = ( x  ^ {1} , x  ^ {2} , x  ^ {3} ) $,
 +
$  dx = dx  ^ {1}  dx  ^ {2}  dx  ^ {3} $
 +
is the volume element of  $  D $,
 +
$  ds $
 +
is the surface element of  $  \Gamma $,
 +
$  N = ( N _ {1} , N _ {2} , N _ {3} ) $
 +
is the unit outer (co-)normal to  $  \Gamma $,
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508021.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508024.png" /> is the volume element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508026.png" /> is the surface element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508028.png" /> is the unit outer (co-)normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508029.png" />,
+
$$
  
<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/g/g045/g045080/g04508030.png" /></td> </tr></table>
+
\frac \partial {\partial  N }
 +
  = \
 +
\sum _ {i = 1 } ^ { 3 }
 +
N _ {i}
 +
\frac \partial {\partial  x  ^ {i} }
  
is the operator of differentiation in the direction of the (co-)vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508031.png" />, and
+
$$
  
<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/g/g045/g045080/g04508032.png" /></td> </tr></table>
+
is the operator of differentiation in the direction of the (co-)vector  $  N $,
 +
and
 +
 
 +
$$
 +
\Delta  = \
 +
\sum _ {i = 1 } ^ { 3 }
 +
\left (
 +
\frac \partial {\partial  x  ^ {i} }
 +
\right )  ^ {2}
 +
$$
  
 
is the [[Laplace operator|Laplace operator]].
 
is the [[Laplace operator|Laplace operator]].
  
Formulas (2) and (3) are also valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508033.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508036.png" /> is the volume element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508038.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508039.png" />-dimensional volume element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508040.png" />, and
+
Formulas (2) and (3) are also valid if $  D $
 +
is a domain in $  E  ^ {n} $,  
 +
$  x = ( x  ^ {1} \dots x  ^ {n} ) $,  
 +
$  dx = dx  ^ {1} \dots dx  ^ {n} $
 +
is the volume element in $  D $,  
 +
$  ds $
 +
is the $  ( n - 1) $-
 +
dimensional volume element of $  \Gamma $,  
 +
and
  
<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/g/g045/g045080/g04508041.png" /></td> </tr></table>
+
$$
 +
\Delta  = \
 +
\sum _ {i = 1 } ^ { n }
 +
\left (
 +
\frac \partial {\partial  x  ^ {i} }
 +
\right )  ^ {2}
 +
$$
  
is the Laplace operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508042.png" /> independent variables.
+
is the Laplace operator in $  n $
 +
independent variables.
  
 
The generalizations of the Green formulas (2) and (3) for linear partial differential operators with sufficiently smooth coefficients have the following form:
 
The generalizations of the Green formulas (2) and (3) for linear partial differential operators with sufficiently smooth coefficients have the following form:
Line 37: Line 139:
 
1) If
 
1) 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/g/g045/g045080/g04508043.png" /></td> </tr></table>
+
$$
 +
Lu  = \
 +
\sum _ {i, j = 1 } ^ { 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/g/g045/g045080/g04508044.png" /></td> </tr></table>
+
\frac \partial {\partial  x  ^ {i} }
  
are (real) adjoint second-order differential operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508045.png" />, then
+
\left [
 +
a  ^ {ij} ( x)
  
<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/g/g045/g045080/g04508046.png" /></td> </tr></table>
+
\frac{\partial  u }{\partial  x  ^ {j} }
  
<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/g/g045/g045080/g04508047.png" /></td> </tr></table>
+
\right ] +
 +
\sum _ {i = 1 } ^ { n }
 +
b  ^ {i} ( x)
  
<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/g/g045/g045080/g04508048.png" /></td> </tr></table>
+
\frac{\partial  u }{\partial  x  ^ {i} }
 +
+
 +
c ( x) u ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508049.png" /> is the unit (co-)vector of the outer normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508050.png" />:
+
$$
 +
L  ^ {*} v  = \sum _ {i, j = 1 } ^ { 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/g/g045/g045080/g04508051.png" /></td> </tr></table>
+
\frac \partial {\partial  x  ^ {i} }
 +
\left [ a  ^ {ij}
 +
( x)
 +
\frac{\partial  v }{\partial  x  ^ {j} }
 +
\right ] - \sum
 +
_ {i = 1 } ^ { n } 
 +
\frac \partial {\partial  x  ^ {i} }
 +
[ b  ^ {i} ( x) v] + c ( x) v
 +
$$
 +
 
 +
are (real) adjoint second-order differential operators,  $  a  ^ {ij} = a  ^ {ji} $,
 +
then
 +
 
 +
$$
 +
\int\limits _ { D }
 +
\left [
 +
vLu +
 +
\sum _ {i, j = 1 } ^ { n }
 +
a  ^ {ij}
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {i} }
 +
 
 +
\frac{\partial  v }{\partial  x  ^ {j} }
 +
-
 +
\left (
 +
\sum _ {i = 1 } ^ { n }
 +
b  ^ {i}
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {i} }
 +
+
 +
cu \right )  v \right ]  dx =
 +
$$
 +
 
 +
$$
 +
= \
 +
\int\limits _  \Gamma  v
 +
\frac{\partial  u }{\partial  M }
 +
  ds,
 +
$$
 +
 
 +
$$
 +
\int\limits _ { D } [ uL  ^ {*} v - vLu]  dx  = \
 +
\int\limits _  \Gamma  \left [ u
 +
\frac{\partial  v }{\partial  M }
 +
- v
 +
 
 +
\frac{\partial  u }{\partial  M }
 +
- B uv \right ]  ds,
 +
$$
 +
 
 +
where  $  N = ( N _ {1} \dots N _ {n} ) $
 +
is the unit (co-)vector of the outer normal to  $  \Gamma $:
 +
 
 +
$$
 +
= \
 +
\sum _ {i = 1 } ^ { n }  N _ {i} b  ^ {i} ,\ \
 +
 
 +
\frac \partial {\partial  M }
 +
  = \
 +
\sum _ {i, j = 1 } ^ { n }
 +
a  ^ {ij} N _ {j}
 +
\frac \partial {\partial  x  ^ {i} }
 +
 
 +
$$
  
 
is the operator of differentiation in the direction of the so-called co-normal
 
is the operator of differentiation in the direction of the so-called co-normal
  
<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/g/g045/g045080/g04508052.png" /></td> </tr></table>
+
$$
 +
= \left (
 +
\sum _ {j = 1 } ^ { n }
 +
a  ^ {1j} N _ {j} \dots
 +
\sum _ {j = 1 } ^ { n }
 +
a  ^ {nj} N _ {j} \right )
 +
$$
  
of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508053.png" />.
+
of the operator $  L $.
  
 
2) If
 
2) 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/g/g045/g045080/g04508054.png" /></td> </tr></table>
+
$$
 +
Lu  = \
 +
\sum _ {i, j = 1 } ^ { n }
 +
a  ^ {ij} ( x)
 +
 
 +
\frac{\partial  ^ {2} u }{\partial  x  ^ {i} \partial  x  ^ {j} }
 +
+
 +
\sum _ {i = 1 } ^ { n }
 +
b  ^ {i} ( x)
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {i} }
 +
+
 +
c ( x) u ,
 +
$$
  
<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/g/g045/g045080/g04508055.png" /></td> </tr></table>
+
$$
 +
L  ^ {*} v  = \sum _ {i, j = 1 } ^ { n } 
 +
\frac{\partial
 +
^ {2} }{\partial  x  ^ {i} \partial  x  ^ {j} }
 +
[ a  ^ {ij} ( x)
 +
v] - \sum _ {i = 1 } ^ { n } 
 +
\frac \partial {\partial  x  ^ {i} }
 +
[ b  ^ {i} ( x) v] + c ( x) v,
 +
$$
  
 
then
 
then
  
<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/g/g045/g045080/g04508056.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { D }
 +
\left [ vLu +
 +
\sum _ {i, j = 1 } ^ { n }
 +
a  ^ {ij}
 +
 
 +
\frac{\partial  u }{\partial  x  ^ {i} }
  
<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/g/g045/g045080/g04508057.png" /></td> </tr></table>
+
\frac{\partial  v }{\partial  x  ^ {j} }
 +
-
 +
\left (
 +
\sum _ {i = 1 } ^ { n }
 +
b  ^ {i}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508058.png" /> is the co-normal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508059.png" />, and
+
\frac{\partial  u }{\partial  x  ^ {i} }
 +
+
 +
cu \right )  v \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/g/g045/g045080/g04508060.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _  \Gamma  v
 +
\frac{\partial  u }{\partial  M }
 +
  ds,
 +
$$
 +
 
 +
where  $  M $
 +
is the co-normal of  $  L $,
 +
and
 +
 
 +
$$
 +
= \
 +
\sum _ {i = 1 } ^ { n }
 +
\left [ b  ^ {i} -
 +
\sum _ {j = 1 } ^ { n }
 +
 
 +
\frac{\partial  a  ^ {ij} }{\partial  x  ^ {j} }
 +
 
 +
\right ] N _ {i} .
 +
$$
  
 
3) If
 
3) 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/g/g045/g045080/g04508061.png" /></td> </tr></table>
+
$$
 +
Lu  = \
 +
\sum _ {p = 1 } ^ { m }  \
 +
\sum _ {| \alpha | = p }
 +
a  ^  \alpha  ( x) D _  \alpha  u + a ( x) u ,
 +
$$
  
<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/g/g045/g045080/g04508062.png" /></td> </tr></table>
+
$$
 +
L  ^ {*} v  = \sum _ {p = 1 } ^ { m }  (- 1)  ^ {p} D _  \alpha  [ a  ^  \alpha  ( x) v] + a ( x) v
 +
$$
  
are (real) adjoint differential operators of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508064.png" /> is an integer multi-index of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508067.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508068.png" />, then
+
are (real) adjoint differential operators of order $  m $,  
 +
$  \alpha = ( \alpha _ {1} \dots \alpha _ {p} ) $
 +
is an integer multi-index of length $  | \alpha | = p $,  
 +
$  1 \leq  \alpha _ {i} \leq  n $,  
 +
$  D _  \alpha  = \partial  _ {\alpha _ {1}  } \dots \partial  _ {\alpha _ {p}  } $,  
 +
and $  \partial  _ {i} = \partial  / \partial  x  ^ {i} $,  
 +
then
  
<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/g/g045/g045080/g04508069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\int\limits _ { D } [ uL  ^ {*} v - vLu]  dx =
 +
$$
  
<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/g/g045/g045080/g04508070.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {p = 1 } ^ { m }  \sum _ {| \alpha |
 +
= p }  \sum _ {k = 1 } ^ { p }  \int\limits _  \Gamma  (- 1)  ^ {k} \left [ \partial  _ {\alpha _ {1}  } \dots \partial  _ {\alpha _ {k - 1 }  } ( a  ^  \alpha  v) \right ] \times
 +
$$
  
<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/g/g045/g045080/g04508071.png" /></td> </tr></table>
+
$$
 +
\times
 +
N _ {\alpha _ {k}  } \left [ \partial  _ {\alpha sub
 +
{k + 1 }  } \dots \partial  _ {\alpha _ {p}  } u \right ]  ds.
 +
$$
  
 
Here the boundary integral can be written as the bilinear sum
 
Here the boundary integral can be written as the bilinear sum
  
<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/g/g045/g045080/g04508072.png" /></td> </tr></table>
+
$$
 +
\sum _ { ij } \int\limits _  \Gamma
 +
B  ^ {ij} ( S _ {i} u) ( T _ {j} v)  ds,
 +
$$
 +
 
 +
where  $  S _ {i} $,
 +
$  T _ {j} $
 +
are certain linear differential operators of orders  $  s _ {i} $,
 +
$  t _ {j} $,
 +
$  0 \leq  s _ {i} + t _ {j} \leq  m - 1 $.
 +
 
 +
Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions  $  u $,
 +
$  v $
 +
which are sufficiently smooth in  $  \overline{D}\; $,
 +
Green's formulas (2) and (4) serve as the source of several relations which are useful in the study of solutions of boundary value problems, in the clarification of the type of boundary value problems, in obtaining explicit solutions, etc. Thus, the function  $  u $
 +
in (2), which is harmonic in  $  D $,
 +
satisfies the [[Gauss theorem|Gauss theorem]]
 +
 
 +
$$
 +
\int\limits _ {\partial  G }
 +
 
 +
\frac{\partial  u }{\partial  N }
 +
\
 +
ds  = 0
 +
$$
 +
 
 +
for  $  v \equiv 1 $.
 +
For sufficiently smooth functions  $  u $,
 +
$  w $
 +
in  $  \overline{D}\; $,
 +
and the function
 +
 
 +
$$
 +
v ( x)  = \
 +
\left \{
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508074.png" /> are certain linear differential operators of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508077.png" />.
+
\begin{array}{ll}
  
Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508079.png" /> which are sufficiently smooth in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508080.png" />, Green's formulas (2) and (4) serve as the source of several relations which are useful in the study of solutions of boundary value problems, in the clarification of the type of boundary value problems, in obtaining explicit solutions, etc. Thus, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508081.png" /> in (2), which is harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508082.png" />, satisfies the [[Gauss theorem|Gauss theorem]]
+
\frac{1}{n - 2 }
 +
| x - y | ^ {2 - n } + w ( x) & \textrm{ if }  n \geq  3, \\
 +
-  \mathop{\rm ln}  | x - y | + w ( x) & \textrm{ if }  n = 2, \\
 +
\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/g/g045/g045080/g04508083.png" /></td> </tr></table>
+
\right .$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508084.png" />. For sufficiently smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508087.png" />, and the function
+
which, for $  x = y $,  
 +
has the same singularity as the fundamental solution of the Laplace operator, the following Green formulas are valid:
  
<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/g/g045/g045080/g04508088.png" /></td> </tr></table>
+
$$ \tag{5 }
 +
\int\limits _ { D }
 +
\left [ u \Delta w +
 +
\sum _ {i = 1 } ^ { n }
  
which, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508089.png" />, has the same singularity as the fundamental solution of the Laplace operator, the following Green formulas are valid:
+
\frac{\partial  u }{\partial  x  ^ {i} }
  
<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/g/g045/g045080/g04508090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\frac{\partial  v }{\partial  x  ^ {i} }
  
<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/g/g045/g045080/g04508091.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
\right ]  dx  = \
 +
cu ( y) +
 +
\int\limits _ { D } u
 +
 
 +
\frac{\partial  v }{\partial  N }
 +
  ds,
 +
$$
 +
 
 +
$$ \tag{6 }
 +
\int\limits _ { D } [ u \Delta w - v \Delta u]  dx  = cu
 +
( y) + \int\limits _ { D } \left [ u
 +
\frac{\partial  v }{\partial
 +
N }
 +
- v
 +
\frac{\partial  u }{\partial  N }
 +
\right ]  ds.
 +
$$
  
 
Here,
 
Here,
  
<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/g/g045/g045080/g04508092.png" /></td> </tr></table>
+
$$
 +
= \left \{
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508093.png" /> is the area of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508094.png" />-dimensional unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508095.png" />. Here it is assumed, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508096.png" />, that the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508097.png" /> has a continuous tangent plane in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g04508098.png" />.
+
\begin{array}{ll}
 +
\omega _ {n}  & \textrm{ if }  y \in D,  \\
 +
{
 +
\frac{1}{2}
 +
} \omega _ {n}  & \textrm{ if }  y \in \Gamma ,  \\
 +
0  & \textrm{ if }  y \notin \overline{D}\; ,  \\
 +
\end{array}
 +
 
 +
\right .$$
 +
 
 +
where $  \omega _ {n} = 2 \pi  ^ {n/2} / \Gamma ( n/2) $
 +
is the area of the $  ( n - 1) $-
 +
dimensional unit sphere in $  E  ^ {n} $.  
 +
Here it is assumed, for $  y \in \Gamma $,  
 +
that the boundary $  \Gamma $
 +
has a continuous tangent plane in some neighbourhood of $  y $.
  
 
Formulas (5) and (6) serve to obtain integral representations of the solutions of basic boundary value problems in [[Potential theory|potential theory]] (cf. [[Harmonic function|Harmonic function]]; [[Green function|Green function]]; [[Poisson formula|Poisson formula]]). Thus, they are used to obtain the Green formula, or Green integral,
 
Formulas (5) and (6) serve to obtain integral representations of the solutions of basic boundary value problems in [[Potential theory|potential theory]] (cf. [[Harmonic function|Harmonic function]]; [[Green function|Green function]]; [[Poisson formula|Poisson formula]]). Thus, they are used to obtain the Green formula, or Green integral,
  
<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/g/g045/g045080/g04508099.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
u ( y)  = \
 +
{
 +
\frac{1}{c}
 +
}
 +
\int\limits _  \Gamma
 +
\left [ v ( x)
 +
 
 +
\frac{\partial  u }{\partial  N }
 +
-
 +
u ( x)
 +
 
 +
\frac{\partial  v }{\partial  N }
 +
 
 +
\right ]  ds,
 +
$$
  
for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g045080100.png" /> which is harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045080/g045080101.png" />. This integral plays an important role in the theory of harmonic functions (cf. [[Harmonic function|Harmonic function]]). Formulas resembling (5) and (6), which give integral representations of the solution of the Cauchy problem or of the mixed problem, are also valid for a normal hyperbolic operator of order two. See [[Kirchhoff formula|Kirchhoff formula]]; [[Riemann method|Riemann method]]; [[Riemann function|Riemann function]]. For Green formulas in the theory of boundary value problems see also [[#References|[4]]]–[[#References|[9]]].
+
for a function $  u $
 +
which is harmonic in $  D $.  
 +
This integral plays an important role in the theory of harmonic functions (cf. [[Harmonic function|Harmonic function]]). Formulas resembling (5) and (6), which give integral representations of the solution of the Cauchy problem or of the mixed problem, are also valid for a normal hyperbolic operator of order two. See [[Kirchhoff formula|Kirchhoff formula]]; [[Riemann method|Riemann method]]; [[Riemann function|Riemann function]]. For Green formulas in the theory of boundary value problems see also [[#References|[4]]]–[[#References|[9]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Green,  "An essay on the application of mathematical analysis to the theories of electricity and magnetism" , Nottingham  (1828)  (Reprint: Mathematical papers, Chelsea, reprint, 1970, pp. 1–82)  {{MR|}}  {{ZBL|21.0014.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Maxwell,  "Selected works on the theory of electromagnetic fields" , Moscow  (1954)  (In Russian; translated from English)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''2''' , Addison-Wesley  (1964)  (Translated from Russian)  {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)  {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</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">[6]</TD> <TD valign="top">  S.L. Sobolev,  "Partial differential equations of mathematical physics" , Pergamon  (1964)  (Translated from Russian)  {{MR|0178220}} {{ZBL|0123.06508}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)  {{MR|0284700}} {{ZBL|0198.14101}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)  {{MR|0188745}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.L. Lions,  E. Magenes,  "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Green,  "An essay on the application of mathematical analysis to the theories of electricity and magnetism" , Nottingham  (1828)  (Reprint: Mathematical papers, Chelsea, reprint, 1970, pp. 1–82)  {{MR|}}  {{ZBL|21.0014.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Maxwell,  "Selected works on the theory of electromagnetic fields" , Moscow  (1954)  (In Russian; translated from English)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''2''' , Addison-Wesley  (1964)  (Translated from Russian)  {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)  {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</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">[6]</TD> <TD valign="top">  S.L. Sobolev,  "Partial differential equations of mathematical physics" , Pergamon  (1964)  (Translated from Russian)  {{MR|0178220}} {{ZBL|0123.06508}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)  {{MR|0284700}} {{ZBL|0198.14101}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)  {{MR|0188745}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.L. Lions,  E. Magenes,  "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)  {{MR|}} {{ZBL|}} </TD></TR></table>

Latest revision as of 19:42, 5 June 2020


Formulas of the integral calculus of functions of several variables, connecting the values of an $ n $- fold integral over a domain $ D $ in an $ n $- dimensional Euclidean space $ E ^ {n} $ with an $ ( n - 1) $- fold integral along the piecewise-smooth boundary $ \partial D = \Gamma $ of this domain. The Green formulas are obtained by integration by parts of integrals of the divergence of a vector field that is continuous in $ \overline{D}\; = D + \Gamma $ and that is continuously differentiable in $ D $.

In the simplest Green formula,

$$ \tag{1 } \int\limits _ \Gamma [ P dx + Q dy] = \ \int\limits \int\limits _ { D } \left [ \frac{\partial Q }{\partial x } - \frac{\partial P }{\partial y } \right ] dx dy, $$

the curvilinear integral along the contour $ \Gamma $ is expressed as a double integral over the domain $ D \subset E ^ {2} $. Here the domain $ D $ is oriented in a natural manner, while an induced orientation, known as counterclockwise, is taken along the boundary $ \Gamma $. Formula (1) has a simple hydrodynamic meaning: The flow across the boundary $ \Gamma $ of a liquid flowing on a plane at rate $ \mathbf v = ( Q, - P) $ is equal to the integral over $ D $ of the intensity (divergence) $ \mathop{\rm div} \mathbf v = ( \partial Q/ \partial x) - ( \partial P/ \partial y) $ of the sources and sinks distributed over $ D $. In this sense the Green formula (1) resembles the Ostrogradski formula (see also Stokes formula).

Formula (1) is sometimes attributed to C.F. Gauss and B. Riemann, but none of its usual appellations corresponds to historical truth; it was in fact encountered as early as the 18th century in the analytical studies of L. Euler and others.

G. Green [1] must be credited with the following formulas of potential theory:

$$ \tag{2 } \int\limits _ { D } \left [ v \Delta u + \sum _ {i = 1 } ^ { 3 } \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {i} } \right ] dx = \ \int\limits _ \Gamma v \frac{\partial u }{\partial N } ds, $$

the preparatory Green formula, and

$$ \tag{3 } \int\limits _ { D } [ u \Delta v - v \Delta u] dx = \ \int\limits _ \Gamma \left [ u \frac{\partial v }{\partial N } - v \frac{\partial u }{\partial N } \right ] ds. $$

Here $ D $ is a domain in $ E ^ {3} $, $ x = ( x ^ {1} , x ^ {2} , x ^ {3} ) $, $ dx = dx ^ {1} dx ^ {2} dx ^ {3} $ is the volume element of $ D $, $ ds $ is the surface element of $ \Gamma $, $ N = ( N _ {1} , N _ {2} , N _ {3} ) $ is the unit outer (co-)normal to $ \Gamma $,

$$ \frac \partial {\partial N } = \ \sum _ {i = 1 } ^ { 3 } N _ {i} \frac \partial {\partial x ^ {i} } $$

is the operator of differentiation in the direction of the (co-)vector $ N $, and

$$ \Delta = \ \sum _ {i = 1 } ^ { 3 } \left ( \frac \partial {\partial x ^ {i} } \right ) ^ {2} $$

is the Laplace operator.

Formulas (2) and (3) are also valid if $ D $ is a domain in $ E ^ {n} $, $ x = ( x ^ {1} \dots x ^ {n} ) $, $ dx = dx ^ {1} \dots dx ^ {n} $ is the volume element in $ D $, $ ds $ is the $ ( n - 1) $- dimensional volume element of $ \Gamma $, and

$$ \Delta = \ \sum _ {i = 1 } ^ { n } \left ( \frac \partial {\partial x ^ {i} } \right ) ^ {2} $$

is the Laplace operator in $ n $ independent variables.

The generalizations of the Green formulas (2) and (3) for linear partial differential operators with sufficiently smooth coefficients have the following form:

1) If

$$ Lu = \ \sum _ {i, j = 1 } ^ { n } \frac \partial {\partial x ^ {i} } \left [ a ^ {ij} ( x) \frac{\partial u }{\partial x ^ {j} } \right ] + \sum _ {i = 1 } ^ { n } b ^ {i} ( x) \frac{\partial u }{\partial x ^ {i} } + c ( x) u , $$

$$ L ^ {*} v = \sum _ {i, j = 1 } ^ { n } \frac \partial {\partial x ^ {i} } \left [ a ^ {ij} ( x) \frac{\partial v }{\partial x ^ {j} } \right ] - \sum _ {i = 1 } ^ { n } \frac \partial {\partial x ^ {i} } [ b ^ {i} ( x) v] + c ( x) v $$

are (real) adjoint second-order differential operators, $ a ^ {ij} = a ^ {ji} $, then

$$ \int\limits _ { D } \left [ vLu + \sum _ {i, j = 1 } ^ { n } a ^ {ij} \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {j} } - \left ( \sum _ {i = 1 } ^ { n } b ^ {i} \frac{\partial u }{\partial x ^ {i} } + cu \right ) v \right ] dx = $$

$$ = \ \int\limits _ \Gamma v \frac{\partial u }{\partial M } ds, $$

$$ \int\limits _ { D } [ uL ^ {*} v - vLu] dx = \ \int\limits _ \Gamma \left [ u \frac{\partial v }{\partial M } - v \frac{\partial u }{\partial M } - B uv \right ] ds, $$

where $ N = ( N _ {1} \dots N _ {n} ) $ is the unit (co-)vector of the outer normal to $ \Gamma $:

$$ B = \ \sum _ {i = 1 } ^ { n } N _ {i} b ^ {i} ,\ \ \frac \partial {\partial M } = \ \sum _ {i, j = 1 } ^ { n } a ^ {ij} N _ {j} \frac \partial {\partial x ^ {i} } $$

is the operator of differentiation in the direction of the so-called co-normal

$$ M = \left ( \sum _ {j = 1 } ^ { n } a ^ {1j} N _ {j} \dots \sum _ {j = 1 } ^ { n } a ^ {nj} N _ {j} \right ) $$

of the operator $ L $.

2) If

$$ Lu = \ \sum _ {i, j = 1 } ^ { n } a ^ {ij} ( x) \frac{\partial ^ {2} u }{\partial x ^ {i} \partial x ^ {j} } + \sum _ {i = 1 } ^ { n } b ^ {i} ( x) \frac{\partial u }{\partial x ^ {i} } + c ( x) u , $$

$$ L ^ {*} v = \sum _ {i, j = 1 } ^ { n } \frac{\partial ^ {2} }{\partial x ^ {i} \partial x ^ {j} } [ a ^ {ij} ( x) v] - \sum _ {i = 1 } ^ { n } \frac \partial {\partial x ^ {i} } [ b ^ {i} ( x) v] + c ( x) v, $$

then

$$ \int\limits _ { D } \left [ vLu + \sum _ {i, j = 1 } ^ { n } a ^ {ij} \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {j} } - \left ( \sum _ {i = 1 } ^ { n } b ^ {i} \frac{\partial u }{\partial x ^ {i} } + cu \right ) v \right ] = $$

$$ = \ \int\limits _ \Gamma v \frac{\partial u }{\partial M } ds, $$

where $ M $ is the co-normal of $ L $, and

$$ C = \ \sum _ {i = 1 } ^ { n } \left [ b ^ {i} - \sum _ {j = 1 } ^ { n } \frac{\partial a ^ {ij} }{\partial x ^ {j} } \right ] N _ {i} . $$

3) If

$$ Lu = \ \sum _ {p = 1 } ^ { m } \ \sum _ {| \alpha | = p } a ^ \alpha ( x) D _ \alpha u + a ( x) u , $$

$$ L ^ {*} v = \sum _ {p = 1 } ^ { m } (- 1) ^ {p} D _ \alpha [ a ^ \alpha ( x) v] + a ( x) v $$

are (real) adjoint differential operators of order $ m $, $ \alpha = ( \alpha _ {1} \dots \alpha _ {p} ) $ is an integer multi-index of length $ | \alpha | = p $, $ 1 \leq \alpha _ {i} \leq n $, $ D _ \alpha = \partial _ {\alpha _ {1} } \dots \partial _ {\alpha _ {p} } $, and $ \partial _ {i} = \partial / \partial x ^ {i} $, then

$$ \tag{4 } \int\limits _ { D } [ uL ^ {*} v - vLu] dx = $$

$$ = \ \sum _ {p = 1 } ^ { m } \sum _ {| \alpha | = p } \sum _ {k = 1 } ^ { p } \int\limits _ \Gamma (- 1) ^ {k} \left [ \partial _ {\alpha _ {1} } \dots \partial _ {\alpha _ {k - 1 } } ( a ^ \alpha v) \right ] \times $$

$$ \times N _ {\alpha _ {k} } \left [ \partial _ {\alpha sub {k + 1 } } \dots \partial _ {\alpha _ {p} } u \right ] ds. $$

Here the boundary integral can be written as the bilinear sum

$$ \sum _ { ij } \int\limits _ \Gamma B ^ {ij} ( S _ {i} u) ( T _ {j} v) ds, $$

where $ S _ {i} $, $ T _ {j} $ are certain linear differential operators of orders $ s _ {i} $, $ t _ {j} $, $ 0 \leq s _ {i} + t _ {j} \leq m - 1 $.

Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions $ u $, $ v $ which are sufficiently smooth in $ \overline{D}\; $, Green's formulas (2) and (4) serve as the source of several relations which are useful in the study of solutions of boundary value problems, in the clarification of the type of boundary value problems, in obtaining explicit solutions, etc. Thus, the function $ u $ in (2), which is harmonic in $ D $, satisfies the Gauss theorem

$$ \int\limits _ {\partial G } \frac{\partial u }{\partial N } \ ds = 0 $$

for $ v \equiv 1 $. For sufficiently smooth functions $ u $, $ w $ in $ \overline{D}\; $, and the function

$$ v ( x) = \ \left \{ \begin{array}{ll} \frac{1}{n - 2 } | x - y | ^ {2 - n } + w ( x) & \textrm{ if } n \geq 3, \\ - \mathop{\rm ln} | x - y | + w ( x) & \textrm{ if } n = 2, \\ \end{array} \right .$$

which, for $ x = y $, has the same singularity as the fundamental solution of the Laplace operator, the following Green formulas are valid:

$$ \tag{5 } \int\limits _ { D } \left [ u \Delta w + \sum _ {i = 1 } ^ { n } \frac{\partial u }{\partial x ^ {i} } \frac{\partial v }{\partial x ^ {i} } \right ] dx = \ cu ( y) + \int\limits _ { D } u \frac{\partial v }{\partial N } ds, $$

$$ \tag{6 } \int\limits _ { D } [ u \Delta w - v \Delta u] dx = cu ( y) + \int\limits _ { D } \left [ u \frac{\partial v }{\partial N } - v \frac{\partial u }{\partial N } \right ] ds. $$

Here,

$$ c = \left \{ \begin{array}{ll} \omega _ {n} & \textrm{ if } y \in D, \\ { \frac{1}{2} } \omega _ {n} & \textrm{ if } y \in \Gamma , \\ 0 & \textrm{ if } y \notin \overline{D}\; , \\ \end{array} \right .$$

where $ \omega _ {n} = 2 \pi ^ {n/2} / \Gamma ( n/2) $ is the area of the $ ( n - 1) $- dimensional unit sphere in $ E ^ {n} $. Here it is assumed, for $ y \in \Gamma $, that the boundary $ \Gamma $ has a continuous tangent plane in some neighbourhood of $ y $.

Formulas (5) and (6) serve to obtain integral representations of the solutions of basic boundary value problems in potential theory (cf. Harmonic function; Green function; Poisson formula). Thus, they are used to obtain the Green formula, or Green integral,

$$ \tag{7 } u ( y) = \ { \frac{1}{c} } \int\limits _ \Gamma \left [ v ( x) \frac{\partial u }{\partial N } - u ( x) \frac{\partial v }{\partial N } \right ] ds, $$

for a function $ u $ which is harmonic in $ D $. This integral plays an important role in the theory of harmonic functions (cf. Harmonic function). Formulas resembling (5) and (6), which give integral representations of the solution of the Cauchy problem or of the mixed problem, are also valid for a normal hyperbolic operator of order two. See Kirchhoff formula; Riemann method; Riemann function. For Green formulas in the theory of boundary value problems see also [4][9].

References

[1] G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism" , Nottingham (1828) (Reprint: Mathematical papers, Chelsea, reprint, 1970, pp. 1–82) Zbl 21.0014.03
[2] J. Maxwell, "Selected works on the theory of electromagnetic fields" , Moscow (1954) (In Russian; translated from English)
[3] V.I. Smirnov, "A course of higher mathematics" , 2 , Addison-Wesley (1964) (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[4] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[5] 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
[6] S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian) MR0178220 Zbl 0123.06508
[7] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[8] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) MR0188745
[9] J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)
How to Cite This Entry:
Green formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Green_formulas&oldid=28207
This article was adapted from an original article by A.K. GushchinL.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article