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A functional connected with the solution of the [[Dirichlet problem|Dirichlet problem]] for the Laplace equation by the variational method. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328701.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328702.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328703.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328704.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328705.png" /> and let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328706.png" /> (cf. [[Sobolev space|Sobolev space]]). The Dirichlet integral for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328707.png" /> is the expression
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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/d/d032/d032870/d0328708.png" /></td> </tr></table>
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For a certain given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328709.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287010.png" /> one considers the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287011.png" /> of functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287012.png" /> which satisfy the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287013.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287014.png" /> is non-empty, there exists a unique function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287015.png" /> for which
+
A functional connected with the solution of the [[Dirichlet problem|Dirichlet problem]] for the Laplace equation by the variational method. Let  $  \Omega $
 +
be a bounded domain in  $  \mathbf R  ^ {n} $
 +
with boundary $  \Gamma $
 +
of class  $  C  ^ {1} $,
 +
let  $  x = ( x _ {1} \dots x _ {n} ) $
 +
and let the function $  u \in W _ {2}  ^ {1} ( \Omega ) $(
 +
cf. [[Sobolev space|Sobolev space]]). The Dirichlet integral for the function  $  u $
 +
is the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287016.png" /></td> </tr></table>
+
$$
 +
D [ u]  = \int\limits _  \Omega  \sum _ {i = 1 } ^ { n }
 +
\left (
 +
\frac{\partial  u }{\partial  x _ {i} }
 +
\right )  ^ {2}  dx .
 +
$$
  
and this function is harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287017.png" />. The converse theorem is also true: If a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287018.png" /> belongs to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287020.png" /> is attained on it. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287021.png" /> is a generalized solution from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287022.png" /> of the Dirichlet problem for the Laplace equation. However, not for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287023.png" /> it is possible to find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287024.png" />. There exists even continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287025.png" /> for which the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287026.png" /> is empty, i.e. the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287027.png" /> contains no functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287028.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287029.png" />. The classical solution of the Dirichlet problem for the Laplace equation with such boundary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287030.png" /> cannot have a finite Dirichlet integral and is not a generalized solution from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287031.png" />.
+
For a certain given function  $  \phi $
 +
on  $  \Gamma $
 +
one considers the set  $  \pi _  \phi  $
 +
of functions from  $  W _ {2}  ^ {1} ( \Omega ) $
 +
which satisfy the boundary condition  $  u \mid  _  \Gamma  = \phi $.
 +
If the set  $  \pi _  \phi  $
 +
is non-empty, there exists a unique function  $  u _ {0} \in \pi _  \phi  $
 +
for which
 +
 
 +
$$
 +
D [ u _ {0} ]  =  \inf _ {u \in \pi _  \phi  }  D [ u] ,
 +
$$
 +
 
 +
and this function is harmonic in $  \Omega $.  
 +
The converse theorem is also true: If a harmonic function $  u _ {0} $
 +
belongs to the set $  \pi _  \phi  $,  
 +
then $  \inf  D [ u] $
 +
is attained on it. Thus, $  u _ {0} $
 +
is a generalized solution from $  W _ {2}  ^ {1} ( \Omega ) $
 +
of the Dirichlet problem for the Laplace equation. However, not for every function $  \phi $
 +
it is possible to find a function $  u _ {0} $.  
 +
There exists even continuous functions on $  \Gamma $
 +
for which the set $  \pi _  \phi  $
 +
is empty, i.e. the space $  W _ {2}  ^ {1} ( \Omega ) $
 +
contains no functions $  u $
 +
satisfying the condition $  u \mid  _  \Gamma  = \phi $.  
 +
The classical solution of the Dirichlet problem for the Laplace equation with such boundary function $  \phi $
 +
cannot have a finite Dirichlet integral and is not a generalized solution from the space $  W _ {2}  ^ {1} ( \Omega ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.P. Mikhailov,  "Partial differential equations" , MIR  (1978)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.P. Mikhailov,  "Partial differential equations" , MIR  (1978)  (Translated from Russian)</TD></TR></table>
  
 +
====Comments====
 +
The restriction of a function (distribution)  $  u $
 +
to a set (in this case the boundary)  $  \Gamma $
 +
is also called the trace of  $  u $
 +
on  $  \Gamma $
 +
in this setting.
  
 +
See [[#References|[a1]]] for a well-known additional reference. Note that the Hilbert space obtained by completion of the set of all  $  C  ^  \infty  $-
 +
functions with compact support with respect to the scalar product
  
====Comments====
+
$$
The restriction of a function (distribution) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287032.png" /> to a set (in this case the boundary) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287033.png" /> is also called the trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287035.png" /> in this setting.
+
( u , v ) \mapsto  \int\limits _  \Omega
 +
\sum _ { i= } 1 ^ { n }
 +
 
 +
\frac{\partial  u }{\partial  x _ {i} }
  
See [[#References|[a1]]] for a well-known additional reference. Note that the Hilbert space obtained by completion of the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287036.png" />-functions with compact support with respect to the scalar product
+
\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/d/d032/d032870/d03287037.png" /></td> </tr></table>
+
$$
  
can be continuously imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287038.png" />. This observation leads to the introduction of the axiomatic theory of Dirichlet spaces, explaining larger parts of classical potential theory (see, e.g., [[#References|[a2]]] or [[#References|[a3]]], and [[Potential theory|Potential theory]]).
+
can be continuously imbedded into $  L  ^ {2} $.  
 +
This observation leads to the introduction of the axiomatic theory of Dirichlet spaces, explaining larger parts of classical potential theory (see, e.g., [[#References|[a2]]] or [[#References|[a3]]], and [[Potential theory|Potential theory]]).
  
 
See also [[Dirichlet principle|Dirichlet principle]]; [[Dirichlet variational problem|Dirichlet variational problem]].
 
See also [[Dirichlet principle|Dirichlet principle]]; [[Dirichlet variational problem|Dirichlet variational problem]].

Revision as of 19:35, 5 June 2020


A functional connected with the solution of the Dirichlet problem for the Laplace equation by the variational method. Let $ \Omega $ be a bounded domain in $ \mathbf R ^ {n} $ with boundary $ \Gamma $ of class $ C ^ {1} $, let $ x = ( x _ {1} \dots x _ {n} ) $ and let the function $ u \in W _ {2} ^ {1} ( \Omega ) $( cf. Sobolev space). The Dirichlet integral for the function $ u $ is the expression

$$ D [ u] = \int\limits _ \Omega \sum _ {i = 1 } ^ { n } \left ( \frac{\partial u }{\partial x _ {i} } \right ) ^ {2} dx . $$

For a certain given function $ \phi $ on $ \Gamma $ one considers the set $ \pi _ \phi $ of functions from $ W _ {2} ^ {1} ( \Omega ) $ which satisfy the boundary condition $ u \mid _ \Gamma = \phi $. If the set $ \pi _ \phi $ is non-empty, there exists a unique function $ u _ {0} \in \pi _ \phi $ for which

$$ D [ u _ {0} ] = \inf _ {u \in \pi _ \phi } D [ u] , $$

and this function is harmonic in $ \Omega $. The converse theorem is also true: If a harmonic function $ u _ {0} $ belongs to the set $ \pi _ \phi $, then $ \inf D [ u] $ is attained on it. Thus, $ u _ {0} $ is a generalized solution from $ W _ {2} ^ {1} ( \Omega ) $ of the Dirichlet problem for the Laplace equation. However, not for every function $ \phi $ it is possible to find a function $ u _ {0} $. There exists even continuous functions on $ \Gamma $ for which the set $ \pi _ \phi $ is empty, i.e. the space $ W _ {2} ^ {1} ( \Omega ) $ contains no functions $ u $ satisfying the condition $ u \mid _ \Gamma = \phi $. The classical solution of the Dirichlet problem for the Laplace equation with such boundary function $ \phi $ cannot have a finite Dirichlet integral and is not a generalized solution from the space $ W _ {2} ^ {1} ( \Omega ) $.

References

[1] V.P. Mikhailov, "Partial differential equations" , MIR (1978) (Translated from Russian)

Comments

The restriction of a function (distribution) $ u $ to a set (in this case the boundary) $ \Gamma $ is also called the trace of $ u $ on $ \Gamma $ in this setting.

See [a1] for a well-known additional reference. Note that the Hilbert space obtained by completion of the set of all $ C ^ \infty $- functions with compact support with respect to the scalar product

$$ ( u , v ) \mapsto \int\limits _ \Omega \sum _ { i= } 1 ^ { n } \frac{\partial u }{\partial x _ {i} } \frac{\partial v }{\partial x _ {i} } $$

can be continuously imbedded into $ L ^ {2} $. This observation leads to the introduction of the axiomatic theory of Dirichlet spaces, explaining larger parts of classical potential theory (see, e.g., [a2] or [a3], and Potential theory).

See also Dirichlet principle; Dirichlet variational problem.

References

[a1] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[a2] J. Deny, "Méthodes Hilbertiennes et théorie du potential" M. Brelot (ed.) H. Bauer (ed.) J.-M. Bony (ed.) J. Deny (ed.) G. Mokobodzki (ed.) , Potential theory (CIME, Stresa, 1969) , Cremonese (1970)
[a3] M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980)
How to Cite This Entry:
Dirichlet integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_integral&oldid=17063
This article was adapted from an original article by A.K. Gushchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article